#612215

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=fg,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 519411);