#480150

order	512
nilpotency class	3
isoclinism	isoclinic class of #17392
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	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×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

965 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)(9,10,13,19,15,20,28,17,14,18,30,35,27,34,12,26)(11,21,24,36,25,38,22,31,23,16,29,32,39,33,40,37);(2,5)(3,7)(9,11)(10,16)(12,22)(13,29)(14,23)(15,25)(17,31)(18,21)(19,36)(20,33)(24,30)(26,37)(27,39)(28,40)(32,35)(34,38);(9,12,27,30,14,28,15,13)(10,17,34,19,18,26,20,35)(11,22,39,24,23,40,25,29)(16,31,38,36,21,37,33,32);(11,23)(16,21)(22,40)(24,29)(25,39)(31,37)(32,36)(33,38)

Polycyclic presentation:

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

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, 387346);