#394121

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#577	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3950 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)(9,10,15,20)(11,21,24,35)(12,17,27,34)(13,19,14,18)(16,28,31,23)(22,37,39,32)(25,40,38,30)(26,36,29,33);(2,4)(9,11)(10,16)(12,25)(13,28)(14,23)(15,24)(17,32)(18,35)(19,21)(20,31)(22,26)(27,38)(29,39)(30,33)(34,37)(36,40);(6,8)(9,12,15,27)(10,17,20,34)(11,22,24,39)(13,29,14,26)(16,30,31,40)(18,36,19,33)(21,37,35,32)(23,38,28,25);(9,11)(10,16)(12,22)(13,23)(14,28)(15,24)(17,30)(18,21)(19,35)(20,31)(25,26)(27,39)(29,38)(32,33)(34,40)(36,37)

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=i,g^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,e^b=ei,g^b=gi,e^c=ei,g^c=gi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301317);