#412208

order	512
nilpotency class	3
isoclinism	isoclinic class of #395151
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 319404);