#6468613

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1575 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)(10,14)(15,16)(17,18,22,23,21,26,20,25)(19,24,29,30,27,31,28,32);(9,10,13,14)(11,16,12,15)(18,23)(20,22)(24,30)(25,26)(28,29)(31,32);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,32)(26,31);(9,12,13,11)(10,16,14,15)(19,27)(24,31)(28,29)(30,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375809);