#395227

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1769 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)(6,7)(9,10)(11,22)(12,28)(13,32)(14,19)(15,21)(16,20)(17,25)(18,31)(23,24)(26,35)(27,36)(29,34)(30,33)(37,40)(38,39);(5,6,8,7)(9,11,14,25)(10,17,19,22)(12,29,30,39)(13,24,18,37)(15,26,16,27)(20,35,21,36)(23,32,40,31)(28,38,33,34);(2,4)(9,12,14,30)(10,18,19,13)(11,23,25,40)(15,32,16,31)(17,34,22,38)(20,33,21,28)(24,36,37,35)(26,29,27,39);(5,7,8,6)(9,13)(10,12)(11,24)(14,18)(15,28)(16,33)(17,29)(19,30)(20,32)(21,31)(22,39)(23,36)(25,37)(26,38)(27,34)(35,40)

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 302423);