#123292

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, prime radical	C₂²	#1619
commutator, Frattini, cosocle	C₈²	C₂³
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 224 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:

(1,2,7,11)(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)(33,34,39,44)(35,45,36,48)(37,51,56,64)(38,57,54,43)(40,49,41,46)(42,60,63,55)(47,62,50,61)(52,59,53,58);(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)(33,35,39,36)(34,40,44,41)(37,52,56,53)(38,50,54,47)(42,61,63,62)(43,59,57,58)(45,64,48,51)(46,60,49,55);(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)(33,36,39,35)(34,41,44,40)(37,53,56,52)(38,47,54,50)(42,62,63,61)(43,58,57,59)(45,51,48,64)(46,55,49,60)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,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, 30488);