#477534

order	512
nilpotency class	3
isoclinism	isoclinic class of #476112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23393	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1069 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:

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

Polycyclic presentation:

2^9|b^2=f,c^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=def,e^b=ei,d^c=de,e^c=ei,h^c=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384730);