#483342

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂²×D₈
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂	C₂²×#335
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

493 conjugacy classes of subgroups; only the 352 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 352 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) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=be,e^a=eg,g^a=gi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320299, #72320313, #72320319, …
maximal normal subgroup of	#72320299, #72320313, #72320319, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390538);