#707419

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1896 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)(3,4)(5,8)(6,7)(11,15)(13,16)(19,23)(22,24)(25,26,28,30)(27,31,32,29);(2,7)(3,5)(18,21)(22,24)(25,26,28,30)(27,29,32,31);(1,3,6,5)(2,8,7,4)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,30)(31,32);(1,4,6,8)(2,3,7,5)(9,11)(10,13)(12,15)(14,16)(17,19,20,23)(18,22,21,24)(25,27,28,32)(26,29,30,31);(1,5,6,3)(2,8,7,4)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=hi,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

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, 614615);