#72263273

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 9956 conjugacy classes of subgroups; only the 3321 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=h,e^2=ij,b^a=bh,c^a=ch,c^b=ch,d^b=dh,e^a=ehj,e^b=eg,e^c=ei,e^d=egij,f^b=fh,f^c=fi,f^d=fj,f^e=fi

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263317
maximal normal subgroup of	#72263317

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