#72332264

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 9579 conjugacy classes of subgroups; only the 3345 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:

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

Polycyclic presentation:

2^10|b^2=hj,d^2=hi,e^2=hj,b^a=bh,c^a=ci,c^b=chij,d^a=dhi,d^b=dij,d^c=di,e^a=ehj,e^c=ei,e^d=eij,f^a=fi,f^c=fi,f^d=fg

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72332285
maximal normal subgroup of	#72332285
maximal quotient of	#72333337

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