#6633268

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632048
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	#25733	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1651 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18)(19,27)(20,31)(21,25)(22,35)(23,29)(24,33)(26,34)(28,43)(30,38)(32,40)(36,39)(37,47)(41,45)(42,46)(44,48);(1,3)(2,5)(4,7)(6,8)(9,10,13,14)(11,16,12,15)(17,19,21,29)(18,23,25,27)(20,28,32,44)(22,30,34,46)(24,37,39,41)(26,38,35,42)(31,48,40,43)(33,45,36,47);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,20,21,32)(18,24,25,39)(19,28,29,44)(22,36,34,33)(23,37,27,41)(26,40,35,31)(30,47,46,45)(38,43,42,48);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(18,25)(23,27)(24,39)(26,35)(31,40)(37,41)(38,42)(43,48);(2,6)(5,8)(18,25)(19,29)(24,39)(26,35)(28,44)(30,46)(31,40)(45,47)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6540464);