#395186

order	512
nilpotency class	3
isoclinism	isoclinic class of #395148
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1739 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^c=fi,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, 302382);