#72258277

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂²⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2463808 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^11|b^a=bg,c^b=ch,d^b=dh,e^a=ei,e^c=eh,f^b=fij,f^e=fk

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72258345
Fitting subgroup of	#72258345, #72258531
maximal subgroup of	#72258296, #72258345, #72265826, …
maximal normal subgroup of	#72258296, #72258345, #72265826, …
maximal quotient of	#72266430

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
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.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);