Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622976 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,11)(8,12)(9,19)(13,21)(14,22)(15,29)(17,25)(18,31)(20,26)(23,24)(27,32)(28,36)(30,35)(33,34);(5,6,12,24,9,15,20,22,10,14,26,29,19,23,8,11)(7,13,25,34,18,28,30,32,16,27,35,36,31,33,17,21);(1,2,3,4)(5,7)(6,13)(8,17)(9,18)(10,16)(11,21)(12,25)(14,27)(15,28)(19,31)(20,30)(22,32)(23,33)(24,34)(26,35)(29,36);(7,16)(13,27)(17,30)(18,31)(21,32)(25,35)(28,33)(34,36);(6,14)(11,22)(13,27)(15,23)(21,32)(24,29)(28,33)(34,36)Polycyclic presentation:
2^9|b^2=fh,c^2=g,f^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fh,h^a=hi,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224826, #72224933, #72224952, … |
| maximal normal subgroup of | #72224826, #72224933, #72224952, … |
| maximal quotient of | #72224818, #72319085, #72319972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530395);