#393722

order	512
nilpotency class	3
isoclinism	isoclinic class of #195963
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1036 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 282 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,26,21,25,27,28)(19,23,30,32,29,20,24,31);(17,19,27,30,21,29,22,24)(18,23,26,31,25,20,28,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,20,21,23)(18,24,25,30)(19,28,29,26)(22,31,27,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,25)(20,23)(26,28)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fg,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,h^b=hi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300918);