#624302

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 531498);