#678222

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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

1804 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)(3,5)(4,8)(6,7)(10,11)(14,15)(17,18,21,20)(19,22,24,23)(25,26)(27,31)(28,29)(30,32);(2,7)(5,8)(9,10,13,11)(12,15,16,14)(18,20)(22,23)(26,29)(30,31);(1,3,6,4)(2,5,7,8)(9,10,13,11)(12,14,16,15)(17,18,21,20)(19,22,24,23)(25,27,28,32)(26,30,29,31);(1,4,6,3)(2,8,7,5)(9,11,13,10)(12,15,16,14)(17,19,21,24)(18,22,20,23);(1,5,6,8)(2,3,7,4)(9,12)(10,14)(11,15)(13,16)(17,20,21,18)(19,23,24,22)(25,26)(27,30)(28,29)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 585418);