#6347392

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347363
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3154 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi

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, 6254588);