Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418488 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,17,20,28)(8,21,22,30)(10,16,11,15)(13,25,26,18)(14,23,27,24)(19,33,35,29)(31,34,36,32);(6,12)(7,18,20,25)(8,22)(13,28,26,17)(15,16)(19,32,35,34)(23,24)(29,36,33,31);(5,7)(6,13)(8,19)(9,20)(10,25)(11,18)(12,26)(14,29)(15,28)(16,17)(21,31)(22,35)(23,34)(24,32)(27,33)(30,36);(2,4)(5,8,11,24,9,22,10,23)(6,14,16,21,12,27,15,30)(7,19,18,32,20,35,25,34)(13,29,17,31,26,33,28,36)Polycyclic presentation:
2^9|a^2=i,b^2=ei,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335262);