Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,15,12,9,11,16)(7,13,19,29,21,18,20,32)(8,22,25,24,27,31,26,14)(17,30,23,33,34,36,35,28);(1,2,3,4)(5,7)(6,13)(8,23)(9,18)(10,19)(11,20)(12,21)(14,30)(15,29)(16,32)(17,26)(22,33)(24,36)(25,34)(27,35)(28,31);(5,8)(6,14)(7,17)(9,24)(10,26)(11,25)(12,27)(13,28)(15,31)(16,22)(18,33)(19,35)(20,23)(21,34)(29,36)(30,32);(5,9,12,6)(7,18,21,13)(8,24,27,14)(10,16,11,15)(17,33,34,28)(19,32,20,29)(22,25,31,26)(23,36,35,30)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302809);