#432254

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×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

1393 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=f,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,e^d=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, 339450);