#424840

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=fi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=dei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332036);