#414401

order	512
nilpotency class	3
isoclinism	isoclinic class of #395151
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

940 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ef,c^2=e,d^2=gh,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, 321597);