#512738

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂	C₂×#2163
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

361 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,f^2=g,g^2=h,h^2=i,b^a=be,c^a=ch,e^a=ei,f^b=fi,d^c=di,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, 419934);