#72267583

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

90273 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)(7,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,27)(22,23)(24,28)(25,26);(6,9)(10,12)(14,18)(19,22)(20,26)(27,28);(13,15)(14,19)(16,24)(17,23)(18,22)(20,28)(21,25)(26,27);(16,25)(20,26)(21,24)(27,28);(6,9)(7,11)(13,15)(14,19)(16,24)(17,23)(18,22)(20,28)(21,25)(26,27);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,16,17,25)(14,20,18,26)(15,21,23,24)(19,27,22,28);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,22)(16,20)(17,18)(19,23)(21,28)(24,27)(25,26)

Polycyclic presentation:

2^10|f^2=hij,b^a=bh,f^a=fi,f^b=fij,f^c=fh,f^d=fh,f^e=fhij,g^a=gj,g^b=gh,g^c=gh,g^e=ghij

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