#72267585

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

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

92805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|b^2=i,f^2=hj,b^a=bh,c^b=chij,d^b=dhij,e^b=ehj,f^a=fij,f^b=fhij,f^c=fhij,f^d=fhij,f^e=fhj,g^a=gj,g^b=ghij,g^c=ghij,g^e=ghj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267594
maximal normal subgroup of	#72267594

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