#405488

order	512
nilpotency class	3
isoclinism	isoclinic class of #394797
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1441 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)(3,5)(4,6)(7,8)(9,10)(11,20)(12,24)(13,28)(14,19)(15,18)(16,22)(17,27)(21,37)(23,33)(25,31)(26,35)(29,32)(30,36)(34,39)(38,40);(1,3,4,7)(2,5,6,8)(9,11,14,29,15,23,13,22)(10,16,28,33,18,32,19,20)(12,25,30,39,27,38,26,21)(17,34,36,31,24,37,35,40);(9,12,13,26,15,27,14,30)(10,17,19,35,18,24,28,36)(11,21,22,38,23,39,29,25)(16,31,20,34,32,40,33,37);(2,6)(5,8)(10,18)(16,32)(17,24)(19,28)(20,33)(31,40)(34,37)(35,36)

Polycyclic presentation:

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

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, 312684);