#508034

order	512
nilpotency class	4
isoclinism	isoclinic class of #508032
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29594
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

966 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 966 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,15)(4,11)(5,10)(6,13)(7,28)(8,14)(9,18)(12,24)(16,31)(17,41)(19,49)(20,42)(21,33)(22,37)(23,39)(25,34)(26,35)(27,38)(29,50)(30,46)(32,48)(36,51)(40,44)(43,57)(45,56)(47,64)(52,60)(53,59)(54,62)(55,61)(58,63);(1,3,7,16,8,18,6,20)(2,9,13,31,14,15,28,42)(4,17,22,43,23,46,27,45)(5,19,24,44,26,48,25,50)(10,32,34,40,35,49,12,29)(11,30,38,57,39,41,37,56)(21,47,51,61,52,63,53,62)(33,58,59,55,60,64,36,54);(2,10)(3,16)(5,24)(6,7)(9,29)(11,33)(12,13)(14,35)(15,40)(17,45)(18,20)(19,48)(21,51)(22,27)(25,26)(28,34)(30,55)(31,49)(32,42)(36,38)(37,59)(39,60)(41,54)(43,46)(52,53)(56,64)(57,58)(61,62);(1,4,8,23)(2,11,14,39)(3,17,18,46)(5,21,26,52)(6,27,7,22)(9,30,15,41)(10,33,35,60)(12,36,34,59)(13,38,28,37)(16,43,20,45)(19,47,48,63)(24,51,25,53)(29,54,40,55)(31,57,42,56)(32,58,49,64)(44,61,50,62)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415230);