#414966

order	512
nilpotency class	3
isoclinism	isoclinic class of #396100
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2414
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15181
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1203 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:

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

Polycyclic presentation:

2^9|a^2=efh,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,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, 322162);