#424820

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1206 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=f,d^2=i,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, 332016);