#803484

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1846 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=fgi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 710680);