#509383

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25441	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10,13,18)(11,21,23,15)(12,26,27,37)(14,20,30,19)(16,31,35,28)(17,24,32,25)(22,29,39,34)(33,36,38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,16)(13,23)(14,24)(17,20)(18,21)(19,32)(22,29)(25,30)(26,28)(27,35)(31,37)(33,36)(34,39)(38,40);(1,3,4,7)(2,5,6,8)(9,12)(10,16)(11,22)(13,27)(14,31)(15,33)(17,34)(18,35)(19,37)(20,26)(21,38)(23,39)(24,40)(25,36)(28,30)(29,32);(2,6)(5,8)(9,11,13,23)(10,17,18,32)(12,22,27,39)(14,25,30,24)(15,19,21,20)(16,34,35,29)(26,33,37,38)(28,40,31,36)

Polycyclic presentation:

2^9|a^2=e,c^2=ei,d^2=i,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, 416579);