#666266

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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

2392 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=gh,e^2=f,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, 573462);