#432255

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	#23422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1449 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)(11,21)(12,17)(13,29)(14,31)(15,20)(16,24)(18,30)(19,22)(23,37)(25,40)(26,35)(27,36)(28,34)(32,39)(33,38);(1,3,5,4)(2,6,8,7)(9,11,22,30,15,13,14,24)(10,16,31,29,20,18,19,21)(12,23,28,39,27,25,26,38)(17,33,35,40,36,32,34,37);(1,3,5,4)(2,7,8,6)(9,12)(10,17)(11,23)(13,25)(14,28)(15,27)(16,32)(18,33)(19,35)(20,36)(21,37)(22,26)(24,39)(29,40)(30,38)(31,34);(1,4,5,3)(2,7,8,6)(9,13,15,11)(10,18,20,16)(12,25,27,23)(14,24,22,30)(17,33,36,32)(19,21,31,29)(26,38,28,39)(34,40,35,37)

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=f,d^2=fi,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, 339451);