#394116

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
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3912 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)(10,16)(12,24)(13,14)(17,29)(19,26)(20,32)(21,36)(22,27)(25,28)(30,33)(35,39)(37,38);(2,4)(9,10)(11,20)(12,18)(13,26)(14,16)(15,19)(17,22)(21,35)(23,29)(24,33)(25,34)(27,32)(28,30)(31,37)(36,40)(38,39);(6,8)(9,11)(10,17)(12,21)(13,27)(14,22)(15,23)(16,29)(18,31)(19,32)(20,26)(24,36)(25,38)(28,37)(30,39)(33,35)(34,40);(9,12,15,25)(10,18,19,34)(11,21,23,38)(13,24,14,28)(16,30,26,33)(17,31,32,40)(20,35,29,39)(22,37,27,36)

Polycyclic presentation:

2^9|d^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, 301312);