#502168

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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

796 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)(13,16)(14,20)(15,17)(18,25)(19,27)(22,37)(23,32)(26,39)(28,36)(29,31)(30,35)(33,40)(34,38);(1,2)(3,4)(5,7,8,6)(10,16)(11,15)(13,27)(17,32)(18,24)(19,21)(20,31)(22,30)(23,29)(25,28)(33,37)(34,39)(35,36)(38,40);(2,4)(6,7)(9,11,14,15)(10,17,20,21)(12,22,26,30)(13,23,29,27)(16,19,31,32)(18,33,34,36)(24,35,39,37)(25,28,38,40);(9,12,14,26)(10,18,20,34)(11,22,15,30)(13,28,29,40)(16,24,31,39)(17,33,21,36)(19,35,32,37)(23,38,27,25)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^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,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409364);