Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13310 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,17)(7,18)(8,16,25,34)(9,13,26,15)(10,27,24,14)(12,28)(19,33)(20,31)(21,30)(22,29)(23,35)(32,36);(5,7)(6,12)(8,21)(9,20)(10,19)(11,23)(13,32)(14,29)(15,30)(16,35)(17,33)(18,34)(22,26)(24,28)(25,36)(27,31);(2,4)(6,13)(7,19)(8,10)(9,26)(12,29)(14,27)(15,17)(18,30)(20,36)(21,22)(23,28)(24,25)(31,33)(32,35)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,f^b=fh,g^b=gi,h^b=hi,d^c=dhi,e^c=ei,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224865, #72224939, #72224952, … |
| maximal normal subgroup of | #72224865, #72224939, #72224952, … |
| maximal quotient of | #72224860, #72237262, #72280205 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55146);