#6583812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574084
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3045	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6491008);