#422936

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1920 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)(9,10)(11,23)(12,28)(13,32)(14,21)(15,20)(16,22)(17,25)(18,24)(19,33)(26,38)(27,37)(29,39)(30,36)(31,40)(34,35);(1,2,3,4)(9,11,14,25,16,27,15,26)(10,17,20,23,22,38,21,37)(12,29,30,34,31,33,24,13)(18,39,40,32,36,19,28,35);(5,6,7,8)(9,12,15,24,16,31,14,30)(10,18,21,28,22,36,20,40)(11,13,25,29,27,34,26,33)(17,35,23,39,38,32,37,19);(2,4)(9,13,16,34)(10,19,22,39)(11,24,27,30)(12,25,31,26)(14,33,15,29)(17,36,38,18)(20,32,21,35)(23,40,37,28)

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=i,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,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^d=ei,h^d=hi

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, 330132);