#496002

order	512
nilpotency class	4
isoclinism	isoclinic class of #33638
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₄×#476
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

528 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=egh,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^c=fhi,g^c=gi,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, 403198);