Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622820 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(6,12)(8,10)(9,21)(13,25)(14,29)(15,26)(17,19)(18,32)(20,23)(22,24)(27,35)(28,36)(30,31)(33,34);(5,6)(7,13)(8,14)(9,22)(10,24)(11,15)(12,21)(16,27)(17,28)(18,33)(19,34)(20,26)(23,29)(25,32)(30,36)(31,35);(2,4)(5,7)(6,13)(8,17)(9,18)(10,19)(11,16)(12,25)(14,28)(15,27)(20,31)(21,32)(22,33)(23,30)(24,34)(26,35)(29,36);(7,16)(13,27)(17,19)(18,30)(25,35)(28,34)(31,32)(33,36);(5,8,11,10)(6,14,15,24)(7,17,16,19)(9,20,23,21)(12,22,26,29)(13,28,27,34)(18,31,30,32)(25,33,35,36)Polycyclic presentation:
2^9|e^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,h^a=hi,f^b=fh,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225127, #72249868, #72249893, … |
| maximal normal subgroup of | #72225127, #72249868, #72249893, … |
| maximal quotient of | #72225017, #72286261, #72287145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530018);