#382687

order	512
nilpotency class	3
isoclinism	isoclinic class of #184792
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2698 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 2698 conjugacy classes of subgroups — too many to draw. Only its 298 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) = 64

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

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

Polycyclic presentation:

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

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, 289883);