#480157

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

961 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,e^2=i,f^2=h,h^2=i,b^a=be,c^a=ci,e^a=eg,d^b=di,e^b=ei,f^b=fgi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387353);