#506262

order	512
nilpotency class	4
isoclinism	isoclinic class of #506261
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#28140
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

570 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)(3,16)(4,11)(5,27)(6,30)(7,12)(8,13)(9,15)(10,19)(14,28)(17,49)(18,36)(20,47)(21,46)(22,35)(23,56)(24,58)(25,41)(26,42)(29,43)(31,38)(32,39)(33,55)(34,52)(37,54)(40,59)(44,60)(45,51)(48,53)(50,62)(57,63)(61,64);(1,3,7,21,6,19,29,52,9,20,28,53,8,22,5,18)(2,10,14,36,30,47,27,46,15,35,12,34,13,16,43,48)(4,23,24,55,32,61,60,51,26,57,59,50,25,54,31,17)(11,37,40,45,39,56,38,62,42,64,58,49,41,63,44,33);(1,4)(2,11)(3,17)(5,24)(6,25)(7,31)(8,32)(9,26)(10,33)(12,38)(13,39)(14,44)(15,42)(16,45)(18,23)(19,50)(20,51)(21,54)(22,55)(27,58)(28,60)(29,59)(30,41)(34,56)(35,62)(36,63)(37,48)(40,43)(46,64)(47,49)(52,57)(53,61);(1,5,8,28,9,29,6,7)(2,12,30,43,15,14,13,27)(3,18,22,53,20,52,19,21)(4,24,32,60,26,59,25,31)(10,34,47,48,35,36,16,46)(11,38,41,40,42,44,39,58)(17,23,55,61,51,57,50,54)(33,56,49,37,62,63,45,64)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413458);