#378946

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=f,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286142);