#376820

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

973 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)(9,10,14,19,15,20,25,18)(11,16,22,31,23,32,12,24)(13,17,28,35,29,36,39,34)(21,30,27,40,37,33,26,38);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,25)(13,21)(14,22)(15,23)(17,30)(18,24)(19,31)(20,32)(26,39)(27,28)(29,37)(33,36)(34,38)(35,40);(9,12,14,11,15,22,25,23)(10,16,18,24,20,32,19,31)(13,27,28,37,29,26,39,21)(17,33,34,40,36,30,35,38);(2,6)(5,7)(9,13,14,28,15,29,25,39)(10,17,19,35,20,36,18,34)(11,21,22,27,23,37,12,26)(16,30,31,40,32,33,24,38)

Polycyclic presentation:

2^9|a^2=h,b^2=e,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^b=gi,d^c=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, 284016);