#508316

order	512
nilpotency class	4
isoclinism	isoclinic class of #508316
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29732
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

680 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415512);