#6490975

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470067
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

3168 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)(9,10,15,20)(11,21,24,34)(12,17,25,31)(13,19,14,18)(16,28,30,23)(22,35,36,40)(26,33,27,32)(29,38,39,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,28)(14,23)(15,24)(17,29)(18,34)(19,21)(20,30)(25,36)(26,38)(27,37)(31,39)(32,40)(33,35);(2,5)(6,8)(9,11)(10,16)(12,22)(13,23)(14,28)(15,24)(17,29)(18,21)(19,34)(20,30)(25,36)(26,37)(27,38)(31,39)(32,35)(33,40);(2,5)(6,8)(9,12)(10,17)(11,22)(13,26)(14,27)(15,25)(16,29)(18,32)(19,33)(20,31)(21,35)(23,37)(24,36)(28,38)(30,39)(34,40);(1,3,4,7)(2,6,5,8)(12,25)(17,31)(22,36)(26,27)(29,39)(32,33)(35,40)(37,38)

Polycyclic presentation:

2^9|a^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,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, 6398171);