#72304532

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 7341 conjugacy classes of subgroups; only the 31843 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:

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

Polycyclic presentation:

2^10|a^2=i,d^2=j,g^2=j,c^b=cij,d^a=di,d^b=di,d^c=dj,e^d=eij,f^d=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, 4, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*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, 4, F.5*F.9*F.10);;
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);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);