#393603

order	512
nilpotency class	3
isoclinism	isoclinic class of #32929
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1580 conjugacy classes of subgroups; only the 402 normal subgroups are shown.

normal subgroups only

This group has 1580 conjugacy classes of subgroups — too many to draw. Only its 402 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=gi,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300799);