#6490958

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398154);