#500603

order	512
nilpotency class	4
isoclinism	isoclinic class of #500602
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24152
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

779 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407799);