#494259

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×#1852
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1017 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:

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

Polycyclic presentation:

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

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, 401455);