#422882

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1979 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:

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

Polycyclic presentation:

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

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, 330078);