Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394020 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2598 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)(7,14)(8,20)(9,15)(10,25)(11,16)(12,18)(13,23)(17,30)(19,28)(21,31)(22,33)(24,32)(26,29)(27,35)(34,36);(1,2,3,4)(5,7)(6,12)(8,21)(9,18)(10,19)(11,26)(13,30)(14,15)(16,28)(17,24)(20,27)(22,35)(23,34)(25,29)(31,33)(32,36);(5,8)(6,13)(7,17)(9,22)(10,24)(11,23)(12,27)(14,31)(15,32)(16,33)(18,34)(19,35)(20,25)(21,26)(28,30)(29,36);(5,7)(6,14)(8,17)(9,18)(10,26)(11,19)(12,15)(13,31)(16,28)(20,36)(21,24)(22,34)(23,35)(25,29)(27,32)(30,33)Polycyclic presentation:
2^9|b^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301422);