#412286

order	512
nilpotency class	3
isoclinism	isoclinic class of #403107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2547
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15129
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1223 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)(3,15,19,33)(4,20,25,37)(5,26)(6,12)(7,13)(8,14)(9,18,34,30)(10,23,38,24)(11,28)(16,31)(17,32,47,58)(21,35)(22,51,54,36)(27,40)(29,39)(41,44)(42,56,55,46)(43,59)(45,48)(49,52,60,53)(50,64)(57,62)(61,63);(1,3)(2,9)(4,21)(5,17)(6,30)(7,18)(8,19)(10,35)(11,32)(12,33)(13,15)(14,34)(16,24)(20,48)(22,50)(23,45)(25,44)(26,55)(27,56)(28,46)(29,47)(31,37)(36,59)(38,41)(39,42)(40,58)(43,53)(49,64)(51,61)(52,63)(54,62)(57,60);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,31)(11,36)(12,20)(13,37)(14,38)(15,41)(17,43)(18,44)(19,45)(21,30)(26,49)(27,52)(28,53)(29,54)(32,57)(33,35)(34,48)(39,60)(40,51)(42,61)(46,62)(47,63)(50,56)(55,59)(58,64);(1,5,7,28,8,29,6,27)(2,11,12,26,14,40,13,39)(3,17,18,46,19,47,30,56)(4,22,24,53,25,54,23,52)(9,32,33,55,34,58,15,42)(10,36,20,49,38,51,37,60)(16,43,44,62,45,63,21,50)(31,57,35,59,48,64,41,61)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319482);