#6485651

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25376	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1545 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,29)(20,25)(21,27)(22,26)(23,28)(24,30)(31,32);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,24)(20,23)(21,30)(22,31)(25,27)(26,29)(28,32);(10,14)(15,16)(17,19,23,20)(18,24,28,32)(21,30,22,31)(25,26,29,27);(9,11,13,12)(10,15,14,16)(17,20,23,19)(18,25,28,29)(21,30,22,31)(24,26,32,27);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,26,28,27)(19,30,20,31)(24,29,32,25)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6392847);