#412292

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

1231 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)(3,14,17,9)(4,19,22,10)(5,24)(6,13)(7,30)(8,12)(11,26)(15,40)(16,45,46,54)(18,34,28,33)(20,48)(21,51,52,55)(23,38,29,37)(25,39)(27,56)(31,43)(32,47,41,57)(35,44)(36,53,49,58)(42,62)(50,59)(60,63)(61,64);(1,3)(2,9)(4,20)(5,16)(6,28)(7,18)(8,17)(10,35)(11,32)(12,14)(13,34)(15,23)(19,48)(21,50)(22,44)(24,54)(25,57)(26,47)(27,46)(29,43)(30,33)(31,37)(36,60)(38,40)(39,41)(42,53)(45,56)(49,62)(51,59)(52,64)(55,61)(58,63);(1,4)(2,10)(3,15)(5,21)(6,29)(7,23)(8,22)(9,31)(11,36)(12,19)(13,38)(14,40)(16,42)(17,43)(18,44)(20,28)(24,55)(25,58)(26,53)(27,52)(30,37)(32,59)(33,48)(34,35)(39,49)(41,61)(45,62)(46,63)(47,64)(50,57)(51,56)(54,60);(1,5,7,26,8,27,6,25)(2,11,13,24,12,39,30,56)(3,16,18,47,17,46,28,57)(4,21,23,53,22,52,29,58)(9,32,34,54,14,41,33,45)(10,36,38,55,19,49,37,51)(15,42,44,64,43,63,20,50)(31,59,35,60,40,61,48,62)

Polycyclic presentation:

2^9|a^2=ef,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,e^c=ei,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, 319488);