#509591

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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×#2215	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

709 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)(9,10,13,16)(11,20,23,32)(12,15,26,28)(14,27,25,19)(17,21,31,37)(18,34,30,22)(24,29,36,35)(33,39,40,38);(5,6)(7,8)(10,15)(11,21)(12,22)(14,25)(16,28)(18,19)(20,35)(23,37)(24,39)(26,34)(27,30)(29,32)(33,40)(36,38);(2,4)(6,8)(10,16)(15,28)(17,31)(18,30)(19,27)(20,32)(29,35)(33,40);(2,4)(9,11,22,38,14,24,26,37,13,23,34,39,25,36,12,21)(10,17,15,29,19,33,30,20,16,31,28,35,27,40,18,32)

Polycyclic presentation:

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

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, 416787);