#845220

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

1690 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=fh,d^2=f,e^2=fg,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, 752416);