#451155

order	512
nilpotency class	3
isoclinism	isoclinic class of #417562
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1620 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)(3,5,7,8)(9,10,14,18,13,19,27,36)(11,16,22,33,21,15,28,31)(12,23,25,34,26,35,29,17)(20,37,38,30,24,40,39,32);(1,3,4,7)(2,5,6,8)(9,11,13,21)(10,15)(12,24)(14,22,27,28)(16,19)(17,32,34,30)(18,31)(20,26)(23,37,35,40)(25,39)(29,38)(33,36);(2,6)(5,8)(9,11)(10,16)(12,20)(13,21)(14,28)(15,19)(17,32)(18,31)(22,27)(23,40)(24,26)(25,39)(29,38)(30,34)(33,36)(35,37);(9,12,13,26)(10,17,19,34)(11,20,21,24)(14,29,27,25)(15,30,16,32)(18,35,36,23)(22,39,28,38)(31,37,33,40)

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,h^c=hi,f^d=fi,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, 358351);