#123294

order	512
nilpotency class	4
isoclinism	isoclinic class of #123275
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1619
commutator, Frattini, cosocle	C₈²	C₂³
prime radical	C₂×D₈	D₈
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

228 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,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,d^b=df,e^b=eg,f^b=fh,g^b=gi,d^c=dg,e^c=eg,f^c=fi,g^c=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, 30490);