#508517

order	512
nilpotency class	4
isoclinism	isoclinic class of #508516
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29912
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

419 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=f,c^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi,g^d=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415713);