#476170

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1309 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383366);