#419572

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×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1604 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,h^d=hi

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, 326768);