#6434365

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1672 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=i,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,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, 6341561);