#381016

order	512
nilpotency class	3
isoclinism	isoclinic class of #195972
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1142 conjugacy classes of subgroups; only the 306 normal subgroups are shown.

normal subgroups only

This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 306 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=ei,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^c=de,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288212);