#403542

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1039 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:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=g,d^2=e,e^2=i,f^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,f^c=fi,h^c=hi,f^d=fi

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, 310738);