#72304547

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

84066 conjugacy classes of subgroups.

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

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^a=ei,e^b=ei,e^c=ej,e^d=eij,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, 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, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*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, 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);