#72304609

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 9657 conjugacy classes of subgroups; only the 31869 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 28 points:

(2,4)(5,6,9,8)(7,10,11,12)(13,14)(15,19)(16,23)(17,18)(20,24)(21,27)(22,25)(26,28);(5,7)(6,10)(8,12)(9,11)(14,17)(16,24)(19,25)(21,28);(6,8)(7,11)(14,17)(16,24)(19,25)(21,28);(2,4)(5,6,9,8)(7,10,11,12)(13,14,18,17)(15,19,22,25)(16,20,24,23)(21,26,28,27);(6,8)(10,12)(13,15)(14,19)(16,21)(17,25)(18,22)(20,26)(23,27)(24,28);(1,2,3,4)(5,8,9,6)(7,12,11,10)(13,16)(14,20)(15,21)(17,23)(18,24)(19,26)(22,28)(25,27);(5,7)(6,10)(8,12)(9,11)(13,17,18,14)(15,19,22,25)(16,20,24,23)(21,27,28,26)

Polycyclic presentation:

2^10|a^2=i,d^2=j,f^2=hj,g^2=ij,b^a=bij,c^a=cj,c^b=ci,d^a=dij,d^b=dij,d^c=dj,e^a=ei,e^d=ei,f^a=fh,f^b=fij,f^c=fj,f^d=fhij,f^e=fi,g^a=gij,g^b=gij,g^c=gj,g^e=gij,g^f=gij

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304735
maximal normal subgroup of	#72304735

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.9);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);