#508038

order	512
nilpotency class	4
isoclinism	isoclinic class of #508036
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

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

normal subgroups only

This group has 958 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fh,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, 415234);