#72339994

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁷×D₄	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

8141208 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:

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

Polycyclic presentation:

2^11|b^2=g,f^2=g,b^a=bg,f^a=fh,f^b=fg,f^c=fi,f^d=fhij,f^e=fgk

Conjugacy classes & character table

392 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339997
maximal normal subgroup of	#72339997

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> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);