#845296

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
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

1742 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)(3,5)(4,6)(7,8)(10,11)(14,15)(25,26,29,28)(27,31,32,30);(1,3,4,7)(2,5,6,8)(9,10,13,11)(12,14,16,15)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,31)(29,32);(2,6)(5,8)(9,11,13,10)(12,14,16,15)(18,21)(22,24)(25,28,29,26)(27,31,32,30);(3,7)(5,8)(9,12,13,16)(10,14,11,15)(26,28)(27,32);(17,19,20,23)(18,22,21,24)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dfg,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, 752492);