Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622820 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
2126 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)(7,17)(8,10)(9,21)(12,25)(14,16)(15,30)(18,34)(19,33)(20,23)(22,24)(26,35)(27,36)(28,29)(31,32);(1,2,3,4)(5,6)(7,12)(8,14)(9,15)(10,16)(11,13)(17,25)(18,27)(19,26)(20,29)(21,30)(22,31)(23,28)(24,32)(33,35)(34,36);(5,7)(6,12)(8,18)(9,22)(10,24)(11,19)(13,26)(14,27)(15,31)(16,32)(17,21)(20,33)(23,34)(25,30)(28,36)(29,35);(6,13)(12,26)(14,16)(15,28)(25,35)(27,32)(29,30)(31,36);(5,8,11,10)(6,14,13,16)(7,18,19,24)(9,20,23,21)(12,27,26,32)(15,29,28,30)(17,22,33,34)(25,31,35,36)Polycyclic presentation:
2^9|b^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=di,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250241, #72250274, #72250328, … |
| maximal normal subgroup of | #72250241, #72250274, #72250328, … |
| maximal quotient of | #72319120, #72320035, C2×#7622938 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530134);