#845144

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

1650 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)(18,20)(22,23)(25,26,29,27)(28,30,32,31);(1,3)(2,5)(4,7)(6,8)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,21)(22,24)(25,27,29,26)(28,31,32,30);(3,7)(5,8)(17,19,21,24)(18,22,20,23)(26,27)(30,31);(10,14)(13,16)(17,20,21,18)(19,22,24,23)(26,27)(30,31);(25,28,29,32)(26,30,27,31)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=f,d^2=f,e^2=g,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, 752340);