#397992

order	512
nilpotency class	3
isoclinism	isoclinic class of #394378
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1201 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=f,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305188);