#624282

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2220 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=fi,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, 531478);