#511762

order	512
nilpotency class	4
isoclinism	isoclinic class of #511761
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32878
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#63090	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3087 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 3087 conjugacy classes of subgroups — too many to draw. Only its 220 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,h^a=hi,e^b=eh,g^b=gi,h^b=hi,f^c=fi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418958);