#397287

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1251 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=ei,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=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, 304483);