#494233

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#120
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1850
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

967 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401429);