#509110

order	512
nilpotency class	4
isoclinism	isoclinic class of #509109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30400
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

644 conjugacy classes of subgroups.

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:

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

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,d^a=def,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=hi

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, 416306);