#480079

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂×#532	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1015 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)(3,7,8,5)(4,6)(9,10,13,19,15,20,29,31,14,18,28,35,30,36,22,38)(11,21,24,37,25,39,12,17,23,16,27,32,40,33,26,34);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,27)(14,23)(15,25)(17,31)(18,21)(19,37)(20,33)(24,28)(26,29)(30,40)(32,35)(34,38)(36,39);(9,12,15,27,14,26,30,24)(10,17,20,32,18,34,36,37)(11,22,25,13,23,29,40,28)(16,31,33,35,21,38,39,19);(11,23)(12,26)(16,21)(17,34)(24,27)(25,40)(32,37)(33,39)

Polycyclic presentation:

2^9|a^2=f,c^2=h,f^2=h,h^2=i,b^a=be,e^a=eg,d^b=di,f^b=fg,d^c=di

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387275);