#502183

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409379);