#72304538

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 9576 conjugacy classes of subgroups; only the 31893 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)(5,6,9,11)(7,10,8,12)(15,21)(19,22)(23,27)(26,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,14,18,17)(15,22,21,19)(16,20,24,25)(23,28,27,26);(1,2)(3,4)(6,11)(10,12)(13,14,18,17)(15,22,21,19)(16,20,24,25)(23,28,27,26);(1,2)(3,4)(5,6,9,11)(7,10,8,12)(13,15,18,21)(14,19,17,22)(16,23,24,27)(20,26,25,28);(5,6)(7,12)(8,10)(9,11)(13,16)(14,20)(15,23)(17,25)(18,24)(19,26)(21,27)(22,28);(2,4)(5,6)(7,12)(8,10)(9,11)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,27)(25,28);(1,2)(3,4)(5,8,9,7)(6,12,11,10)(13,17,18,14)(15,22,21,19)(16,20,24,25)(23,26,27,28)

Polycyclic presentation:

2^10|a^2=i,b^2=j,c^2=ij,d^2=j,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=fh,f^d=fh,g^e=gj,g^f=ghij

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