#395230

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1721 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)(11,17)(12,27)(13,29)(14,20)(15,19)(16,21)(18,31)(22,38)(23,39)(24,36)(25,35)(26,37)(28,33)(30,32)(34,40);(5,6,8,7)(9,11,16,26)(10,17,21,37)(12,28,18,34)(13,23,32,38)(14,24,15,25)(19,35,20,36)(22,29,39,30)(27,33,31,40);(2,4)(9,12,16,18)(10,13,21,32)(11,22,26,39)(14,30,15,29)(17,33,37,40)(19,31,20,27)(23,36,38,35)(24,34,25,28);(5,7,8,6)(9,13)(10,18)(11,23)(12,21)(14,31)(15,27)(16,32)(17,34)(19,29)(20,30)(22,36)(24,40)(25,33)(26,38)(28,37)(35,39)

Polycyclic presentation:

2^9|b^2=ei,c^2=i,d^2=e,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, 302426);