#6631528

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631114
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1454 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,d^b=di,d^c=dfh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538724);