#72304562

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 9603 conjugacy classes of subgroups; only the 31845 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=j,c^2=ij,d^2=j,f^2=hj,g^2=j,c^a=ci,d^a=dij,d^b=dij,d^c=dj,e^a=ei,e^b=ei,e^c=ei,e^d=ei,f^a=fh,f^b=fh,f^c=fhi,f^d=fhij,f^e=fi,g^e=gj,g^f=gh

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, 2, F.10);;
gap> SetPower(coll, 3, F.9*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, 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, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, 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.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
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, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);