#455241

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1301 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)(9,10,15,20,13,18,14,19)(11,21,31,33,24,16,25,22)(12,26,30,17,28,34,29,27)(23,37,40,35,38,36,39,32);(5,6,8,7)(9,11,13,24)(10,16,18,21)(12,23,28,38)(14,25,15,31)(17,32,27,35)(19,33,20,22)(26,36,34,37)(29,39,30,40);(2,4)(5,6)(7,8)(9,10,13,18)(11,22,24,33)(12,27,28,17)(14,20,15,19)(16,25,21,31)(23,37,38,36)(26,30,34,29)(32,39,35,40);(5,7,8,6)(9,12)(10,17)(11,23)(13,28)(14,30)(15,29)(16,32)(18,27)(19,34)(20,26)(21,35)(22,36)(24,38)(25,40)(31,39)(33,37)

Polycyclic presentation:

2^9|a^2=ef,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,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362437);