#452172

order	512
nilpotency class	3
isoclinism	isoclinic class of #417566
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1162 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:

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

Polycyclic presentation:

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

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, 359368);