#9937832

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9845028);