#413064

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1602 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=e,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,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, 320260);