#122866

order	512
nilpotency class	4
isoclinism	isoclinic class of #122865
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#331
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₈²	C₂³
prime radical	C₂³	#331
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

256 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,4,9,5,8,17,24,6,11,18,27,20,10,28,26)(3,12,13,29,14,23,30,32,15,22,31,25,19,7,16,21)(33,34)(35,45,47,61)(36,50)(37,44)(38,43)(39,41)(40,46,55,63)(42,52)(48,59,49,62)(51,57,64,56)(53,58)(54,60);(1,3,6,15)(2,7,11,23)(4,16,18,30)(5,19,20,14)(8,22,10,12)(9,25,27,29)(13,17,31,28)(21,24,32,26)(33,35,39,47)(34,40,41,55)(36,51,54,64)(37,49,38,48)(42,59,58,62)(43,57,44,56)(45,50,61,60)(46,52,63,53);(34,41)(40,55)(42,58)(43,44)(45,61)(50,60)(56,57)(59,62)

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=gi,e^2=fg,f^2=h,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,g^b=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30062);