#123372

order	512
nilpotency class	4
isoclinism	isoclinic class of #123363
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	#1667
commutator, Frattini, cosocle	C₈²	C₂³
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

147 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 147 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=df,c^2=i,d^2=fh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi,e^c=eg,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30568);