#508033

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
prime radical	#60471	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 965 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)(4,20)(5,26)(6,12)(7,31)(8,13)(9,16)(10,22)(11,29)(15,44)(17,49)(18,43)(19,34)(21,56)(23,38)(24,60)(25,55)(27,40)(28,39)(30,54)(32,62)(33,51)(35,50)(36,58)(37,59)(41,46)(42,52)(45,63)(47,53)(48,61)(57,64);(1,3,7,19,8,16,6,18)(2,9,12,34,13,14,31,43)(4,21,24,15,25,47,23,45)(5,17,29,52,30,48,28,51)(10,35,36,41,37,64,40,32)(11,33,26,61,39,42,54,49)(20,53,38,44,55,56,60,63)(22,57,27,46,59,50,58,62);(1,4,8,25)(2,10,13,37)(3,15,16,45)(5,27,30,58)(6,24,7,23)(9,32,14,41)(11,38,39,60)(12,40,31,36)(17,50,48,57)(18,47,19,21)(20,54,55,26)(22,28,59,29)(33,53,42,56)(34,64,43,35)(44,61,63,49)(46,51,62,52);(1,2)(3,9)(4,10)(5,11)(6,31)(7,12)(8,13)(14,16)(15,41)(17,33)(18,43)(19,34)(20,22)(21,35)(23,40)(24,36)(25,37)(26,29)(27,38)(28,54)(30,39)(32,45)(42,48)(44,46)(47,64)(49,51)(50,56)(52,61)(53,57)(55,59)(58,60)(62,63)

Polycyclic presentation:

2^9|b^2=h,c^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, 415229);