Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1446 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,15,18,28)(8,12)(9,13)(10,14)(11,17,24,21)(16,31,33,30)(19,26)(20,27)(22,29)(23,34,25,32)(35,36);(5,7)(6,11)(8,16,20,33)(9,21)(10,18)(12,25,27,23)(13,28)(14,24)(15,29)(17,22)(19,34,35,32)(26,30,36,31);(1,2,3,4)(5,8)(6,12)(7,16,18,33)(9,19)(10,20)(11,23,24,25)(13,26)(14,27)(15,30,28,31)(17,32,21,34)(22,35)(29,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72279381, #72279624, #72279907, … |
| maximal normal subgroup of | #72279381, #72279624, #72279907, … |
| maximal quotient of | #72280976, #72281202, C2×#124041 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31237);