#494317

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1862
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1021 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,h^a=hi,e^b=eh,f^b=fhi,h^b=hi,e^c=ehi,f^c=fhi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401513);