#493894

order	512
nilpotency class	4
isoclinism	isoclinic class of #480315
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×#1820
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ #3428 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

837 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)(3,13,15,34)(4,18)(5,10)(6,24,25,11)(7,12)(8,16,27,40)(9,21,29,48)(14,36,39,60)(17,43)(19,45)(20,46)(22,50,51,31)(23,32)(26,53)(28,42)(30,49,55,63)(33,56)(35,41)(37,59,61,44)(38,57)(47,58)(52,62)(54,64);(1,3)(2,8,12,27)(4,14)(5,16)(6,17)(7,15)(9,26,29,54)(10,13,32,34)(11,28,24,52)(18,44,45,59)(19,39)(20,37)(21,38)(22,42)(23,40)(25,41)(30,33,55,57)(31,35,50,43)(36,58,60,46)(47,61)(48,56)(49,53)(51,62)(63,64);(1,4,5,20,7,19,23,47)(2,9,10,30,12,29,32,55)(3,14,16,37,15,39,40,61)(6,21,22,49,25,48,51,63)(8,26,13,33,27,54,34,57)(11,18,31,46,24,45,50,58)(17,38,42,53,41,56,62,64)(28,44,35,36,52,59,43,60);(3,15)(4,19)(8,27)(9,29)(13,34)(16,40)(17,41)(18,45)(20,47)(21,48)(28,52)(30,55)(35,43)(42,62)(46,58)(49,63)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,e^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,g^b=gi,d^c=di,g^c=gh,g^f=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, 401090);