#509288

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	#28247	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

545 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)(3,4)(5,7)(6,8)(9,10)(11,14)(12,25)(13,23)(15,19)(16,20)(17,26)(18,22)(21,27)(24,33)(28,35)(29,36)(30,31)(32,37)(34,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,17)(13,22)(15,30)(16,24)(18,23)(19,31)(20,33)(21,27)(25,26)(28,36)(29,35)(32,34)(37,39)(38,40);(1,4,5,6)(2,3,7,8)(9,12,16,29)(10,17,20,35)(11,21,24,38)(13,28,15,26)(14,32,33,39)(18,27,31,40)(19,36,23,25)(22,34,30,37);(2,7)(3,8)(9,11)(10,18)(12,21)(13,30)(14,23)(15,22)(16,24)(17,27)(19,33)(20,31)(25,32)(26,34)(28,37)(29,38)(35,40)(36,39)

Polycyclic presentation:

2^9|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, 416484);