#509652

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	#13336	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 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:

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

Polycyclic presentation:

2^9|a^2=e,d^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, 416848);