#420769

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1916 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)(9,10)(11,21)(12,24)(13,18)(14,31)(15,20)(16,23)(17,27)(19,29)(22,33)(25,38)(26,40)(28,37)(30,34)(32,36)(35,39);(1,2)(3,4)(5,6)(7,8)(9,11,14,23,15,13,29,30)(10,16,19,21,20,34,31,18)(12,25,27,26,28,39,32,22)(17,35,24,40,36,38,37,33);(6,8)(9,12)(10,17)(11,22)(13,26)(14,32)(15,28)(16,33)(18,35)(19,37)(20,36)(21,38)(23,39)(24,31)(25,30)(27,29)(34,40);(2,4)(9,13,15,11)(10,18,20,21)(12,26,28,22)(14,23,29,30)(16,19,34,31)(17,35,36,38)(24,33,37,40)(25,32,39,27)

Polycyclic presentation:

2^9|b^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 327965);