#72256663

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 6863 conjugacy classes of subgroups; only the 31813 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:

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

Polycyclic presentation:

2^10|b^2=j,c^2=ij,d^2=h,e^2=j,b^a=bj,c^a=ci,d^a=dh,d^c=dhij,e^d=eij,f^c=fij,f^e=fj,g^c=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256666, #72256702
maximal normal subgroup of	#72256666, #72256702

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