#424819

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1270 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332015);