#494875

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1925
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

761 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,e^2=i,f^2=hi,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,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, 402071);