#845274

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

1654 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)(11,13)(12,14)(15,16)(17,18,21,19)(20,23,24,22)(25,26)(27,31)(28,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,19,21,18)(20,23,24,22)(26,29)(30,31);(2,5)(6,8)(17,20,21,24)(18,22,19,23)(26,29)(27,32);(10,14)(13,16)(18,19)(22,23)(25,27,28,32)(26,30,29,31);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fg,d^2=f,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, 752470);