Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417601 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,17)(8,23)(9,14)(10,16)(11,15)(12,20)(13,26)(18,29)(19,35)(21,30)(22,31)(24,28)(25,34)(27,33)(32,36);(1,2)(3,4)(5,7)(6,12)(8,24)(9,20)(10,21)(11,22)(13,32)(14,17)(15,30)(16,31)(18,26)(19,25)(23,35)(27,36)(28,34)(29,33);(5,8)(6,13)(7,18)(9,25)(10,27)(11,26)(12,28)(14,33)(15,34)(16,23)(17,35)(19,22)(20,36)(21,24)(29,30)(31,32);(1,3,4,2)(5,8)(6,13)(7,19)(9,25)(10,26)(11,27)(12,29)(14,33)(15,23)(16,34)(17,32)(18,22)(20,24)(21,36)(28,30)(31,35)Polycyclic presentation:
2^9|d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325561);