#403568

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
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:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=g,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^c=fi,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, 310764);