#6475681

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469995
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	#35261	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6382877);