#6792484

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37656	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6699680);