#501685

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	#24778	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

684 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,d^2=eh,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,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408881);