#72329178

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

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 9808 conjugacy classes of subgroups; only the 3325 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:

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

Polycyclic presentation:

2^10|b^2=gh,e^2=hij,f^2=gij,b^a=bg,e^a=egi,e^b=ej,e^c=egi,e^d=ehij,f^a=fhi,f^b=fj,f^c=fij,f^d=fgij,f^e=fgj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329191
maximal normal subgroup of	#72329191

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