#6407970

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#52307	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1700 conjugacy classes of subgroups; only the 746 normal subgroups are shown.

normal subgroups only

This group has 1700 conjugacy classes of subgroups — too many to draw. Only its 746 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 — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=df,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6315166);