#6460055

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444408
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

2188 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)(9,10,13,19,15,20,14,17)(11,21,22,30,25,32,24,16)(12,18,28,35,29,36,27,33)(23,37,38,40,39,34,26,31);(2,5)(6,8)(9,11)(10,16)(12,26)(13,24)(14,22)(15,25)(17,21)(18,34)(19,32)(20,30)(23,27)(28,39)(29,38)(31,33)(35,40)(36,37);(10,17,20,19)(11,22,25,24)(16,30)(18,33,36,35)(21,32)(23,38,39,26)(31,40)(34,37);(1,3,4,7)(2,6,5,8)(9,12,13,28,15,29,14,27)(10,18,19,35,20,36,17,33)(11,23,22,38,25,39,24,26)(16,31,21,37,30,40,32,34);(9,13,15,14)(10,19,20,17)(11,24,25,22)(12,27,29,28)(16,32,30,21)(18,33,36,35)(23,38,39,26)(31,37,40,34)

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=f,e^2=i,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^d=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, 6367251);