#72227002

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈² · Schur multiplier (H₂) = C₂⁷×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₈²	C₂⁵
commutator, socle	C₂²	C₂³×C₈²
prime radical	C₂²×#41	C₄×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

5800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

(17,18)(19,24)(20,32)(21,26)(22,25)(23,27)(28,41)(29,37)(30,38)(31,36)(33,45)(34,39)(35,40)(42,48)(43,47)(44,46);(1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16);(17,19,21,30,22,31,34,44,23,29,35,43,20,28,33,42)(18,24,26,38,25,36,39,46,27,37,40,47,32,41,45,48);(17,20,23,22)(18,25,27,32)(19,28,29,31)(21,33,35,34)(24,36,37,41)(26,39,40,45)(30,42,43,44)(38,46,47,48);(2,6)(7,12)(8,13)(14,16)(19,29)(24,37)(28,31)(30,43)(36,41)(38,47)(42,44)(46,48)

Polycyclic presentation:

2^11|b^2=f,c^2=g,d^2=k,f^2=h,g^2=i,h^2=j,i^2=k,d^a=dk,e^b=ej,e^c=ek

Conjugacy classes & character table

704 conjugacy classes — view the full character table.

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