#72328002

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 9978 conjugacy classes of subgroups; only the 3373 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^10|c^2=h,e^2=g,f^2=hi,b^a=bj,c^a=ci,c^b=cg,d^c=dh,e^a=ei,e^b=eh,e^c=egh,e^d=eh,f^d=fhi,f^e=fi

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328076
maximal normal subgroup of	#72328076
maximal quotient of	#72331237

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