#509657

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,4,5)(3,7,8,6)(9,10,14,20)(11,24,27,17)(12,31,34,52)(13,16,22,41)(15,23,40,21)(18,42,50,35)(19,38,53,39)(25,58,60,70)(26,43,37,63)(28,47,30,55)(29,57,46,48)(32,67,68,65)(33,59,54,36)(44,51,66,71)(45,61,56,62)(49,69,72,64);(2,5)(3,8)(9,11,14,27)(10,17,20,24)(12,32,34,68)(13,37,22,26)(15,28,40,30)(16,43,41,63)(18,49,50,72)(19,29,53,46)(21,47,23,55)(25,59,60,36)(31,65,52,67)(33,70,54,58)(35,69,42,64)(38,48,39,57)(44,61,66,62)(45,71,56,51);(9,12,14,34)(10,18,20,50)(11,25,27,60)(13,33,22,54)(15,42,40,35)(16,44,41,66)(17,45,24,56)(19,36,53,59)(21,52,23,31)(26,49,37,72)(28,62,30,61)(29,64,46,69)(32,63,68,43)(38,51,39,71)(47,58,55,70)(48,65,57,67);(1,3,4,8)(2,6,5,7)(9,13,14,22)(10,19,20,53)(11,26,27,37)(12,33,34,54)(15,39,40,38)(16,23,41,21)(17,46,24,29)(18,36,50,59)(25,49,60,72)(28,48,30,57)(31,66,52,44)(32,58,68,70)(35,51,42,71)(43,47,63,55)(45,69,56,64)(61,67,62,65)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416853);