#509542

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	#3058	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

466 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ei,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, 416738);