#378947

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
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

800 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)(9,10,16,21,14,20,15,22)(11,23,13,32,25,17,26,19)(12,27,31,35,29,36,30,18)(24,38,28,33,39,34,40,37);(1,3,4,8)(2,5,6,7)(9,11,16,13,14,25,15,26)(10,17,21,19,20,23,22,32)(12,28,30,24,29,40,31,39)(18,34,36,33,35,38,27,37);(9,12,15,30,14,29,16,31)(10,18,21,27,20,35,22,36)(11,24,26,40,25,39,13,28)(17,33,19,34,23,37,32,38);(2,6)(5,7)(9,13,16,25,14,26,15,11)(10,19,21,23,20,32,22,17)(12,28,31,39,29,40,30,24)(18,34,27,37,35,38,36,33)

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,d^b=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286143);