#845205

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

1602 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=h,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, 752401);