#6460047

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367243);