#509860

order	512
nilpotency class	4
isoclinism	isoclinic class of #509858
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31064
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24285	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

804 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417056);