#508036

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

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

normal subgroups only

This group has 971 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)(3,14)(4,10)(5,9)(6,25)(7,13)(8,15)(11,22)(12,23)(16,37)(17,26)(18,29)(19,31)(20,35)(21,50)(24,33)(27,40)(28,45)(30,41)(32,51)(34,47)(36,48)(38,46)(39,42)(43,52)(44,55)(49,56)(53,60)(54,63)(57,64)(58,62)(59,61);(1,3,6,18,7,15,23,42)(2,8,12,29,13,14,25,39)(4,16,21,44,20,40,48,61)(5,17,22,46,24,41,51,45)(9,30,32,38,33,26,11,28)(10,27,36,55,35,37,50,59)(19,43,47,62,49,60,64,63)(31,53,57,58,56,52,34,54);(2,9)(3,15)(5,22)(6,23)(8,26)(10,31)(11,12)(13,33)(14,30)(17,45)(19,47)(21,48)(24,51)(25,32)(27,53)(28,39)(29,38)(34,36)(35,56)(37,52)(41,46)(43,62)(44,61)(49,64)(50,57)(54,55)(58,59)(60,63);(1,4)(2,10)(3,16)(5,19)(6,21)(7,20)(8,27)(9,31)(11,34)(12,36)(13,35)(14,37)(15,40)(17,43)(18,44)(22,47)(23,48)(24,49)(25,50)(26,52)(28,54)(29,55)(30,53)(32,57)(33,56)(38,58)(39,59)(41,60)(42,61)(45,63)(46,62)(51,64)

Polycyclic presentation:

2^9|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, 415232);