#72304554

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 7488 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:

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

Polycyclic presentation:

2^10|a^2=i,c^2=i,d^2=j,g^2=j,c^a=ci,c^b=cj,d^a=di,d^b=di,d^c=dj,e^a=ei,e^b=ei,e^c=ei,e^d=eij,f^c=fh,f^d=fh,f^e=fh,g^e=gj,g^f=ghi

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