#72300533

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 10148 conjugacy classes of subgroups; only the 3421 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)(5,6,9,12)(7,8,11,10)(21,22)(23,24)(25,26)(27,28);(5,6)(7,8)(9,12)(10,11)(14,17)(18,19)(22,26)(23,27);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(13,14,16,17)(15,19,20,18)(21,23,25,27)(22,24,26,28);(7,11)(8,10)(15,20)(18,19)(23,27)(24,28);(5,7)(6,8)(9,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,24)(22,23)(25,28)(26,27);(5,8)(6,11)(7,12)(9,10)(13,14)(15,19)(16,17)(18,20)(21,22)(23,28)(24,27)(25,26)

Polycyclic presentation:

2^10|a^2=h,c^2=gij,b^a=bi,c^a=cg,c^b=cgij,d^c=di,e^c=egij,e^d=egij,f^b=fgij,f^c=fgij,f^d=fh,f^e=fgij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300575, #72326545
maximal normal subgroup of	#72300575, #72326545

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