#72286177

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

89777 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)(5,6)(7,9)(8,11)(10,12)(15,21)(18,25)(22,24)(26,28);(5,7,8,12)(6,9,11,10)(13,14,17,20)(15,18,21,25)(16,19,23,27)(22,26,24,28);(5,7)(6,10)(8,12)(9,11)(16,23)(19,27)(22,24)(26,28);(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,27)(24,28);(1,2)(3,4)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,27)(24,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,24)(17,21)(19,28)(20,25)(22,23)(26,27);(5,6)(7,10)(8,11)(9,12)(13,16)(14,19)(15,22)(17,23)(18,26)(20,27)(21,24)(25,28)

Polycyclic presentation:

2^10|b^2=i,c^a=cij,c^b=cij,d^b=dj,e^a=eh,e^b=ej,f^a=fi,f^b=fij,g^b=gij,g^c=gj,g^f=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286214, #72286795, #72286876
maximal normal subgroup of	#72286214, #72286795, #72286876

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