#407053

order	512
nilpotency class	3
isoclinism	isoclinic class of #393899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1227 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)(11,15)(12,25,27,35)(13,18)(14,17)(16,28,34,29)(19,20)(21,33,37,40)(22,32)(23,36)(24,31)(26,30,38,39);(1,3,4,7)(2,5,6,8)(9,11,20,23,14,24,13,22)(10,15,19,36,17,31,18,32)(12,26,28,21,27,38,29,37)(16,33,25,39,34,40,35,30);(9,12,13,29,14,27,20,28)(10,16,18,35,17,34,19,25)(11,21,22,26,24,37,23,38)(15,30,32,40,31,39,36,33);(2,6)(5,8)(10,17)(15,31)(16,34)(18,19)(25,35)(30,39)(32,36)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314249);