#419348

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1586 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)(17,18,21,25)(19,26,22,23)(20,28,30,31)(24,29,32,27);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,22)(18,23,25,26)(20,29,30,27)(24,31,32,28);(2,4)(6,8)(9,12,13,10)(11,15,16,14);(11,16)(14,15)(17,20,21,30)(18,24,25,32)(19,27,22,29)(23,28,26,31)

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=ei,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, 326544);