#415617

order	512
nilpotency class	3
isoclinism	isoclinic class of #394043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2613
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15203
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1155 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)(3,15,19,9)(4,20,25,36)(5,11)(6,13)(7,14)(8,12)(10,23,35,24)(16,43)(17,31,46,42)(18,40,29,32)(21,30)(22,52,53,50)(26,38)(27,37)(28,39)(33,45)(34,54,60,55)(41,49)(44,62)(47,57,48,58)(51,56)(59,63)(61,64);(1,3)(2,9)(4,21)(5,17)(6,18)(7,29)(8,19)(10,33)(11,31)(12,15)(13,32)(14,40)(16,24)(20,41)(22,51)(23,45)(25,49)(26,46)(27,47)(28,48)(30,36)(34,59)(35,43)(37,57)(38,42)(39,58)(44,54)(50,61)(52,56)(53,64)(55,63)(60,62);(1,4)(2,10)(3,16)(5,22)(6,24)(7,23)(8,25)(9,30)(11,34)(12,35)(13,36)(14,20)(15,41)(17,44)(18,49)(19,45)(21,29)(26,53)(27,55)(28,54)(31,56)(32,43)(33,40)(37,50)(38,60)(39,52)(42,61)(46,63)(47,51)(48,64)(57,59)(58,62);(1,5,7,27,8,26,6,28)(2,11,14,37,12,38,13,39)(3,17,29,47,19,46,18,48)(4,22,23,55,25,53,24,54)(9,31,40,57,15,42,32,58)(10,34,20,50,35,60,36,52)(16,44,21,51,45,63,49,64)(30,56,33,59,41,61,43,62)

Polycyclic presentation:

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

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, 322813);