#122494

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₈²	C₂³
socle	C₂²	C₂×D₄⋊C₈
prime radical	#466	C₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

247 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,12)(4,9)(5,17)(6,21)(7,11)(8,14)(10,19)(13,28)(15,29)(16,27)(18,23)(20,31)(22,26)(24,32)(25,30)(33,34,36,41,38,42,49,56,37,40,48,53,50,54,61,63)(35,43,45,55,46,57,59,64,44,39,47,51,58,52,60,62);(1,3)(2,8)(4,13)(5,18)(6,22)(7,15)(9,23)(10,26)(11,25)(12,17)(14,19)(16,30)(20,32)(21,24)(27,28)(29,31)(33,35)(34,39)(36,47)(37,44)(38,46)(40,43)(41,55)(42,52)(45,48)(49,60)(50,58)(51,53)(54,57)(56,64)(59,61)(62,63);(1,4,7,6)(2,9,11,21)(3,13,15,22)(5,16,20,19)(8,23,25,24)(10,17,27,31)(12,28,29,26)(14,18,30,32)(35,44)(39,43)(45,47)(46,58)(51,55)(52,57)(59,60)(62,64)

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=fhi,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ci,d^b=dfi,e^b=ei,f^b=fh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29690);