Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,18)(8,14)(9,12)(10,16)(11,17)(13,21)(15,24)(19,31)(20,32)(22,33)(23,28)(25,30)(26,29)(27,35)(34,36);(1,2)(3,4)(6,12)(8,10)(9,24)(13,27)(14,15)(16,28)(17,29)(18,32)(19,21)(20,36)(23,26)(25,35)(30,33)(31,34);(2,4)(5,7)(6,13)(8,19)(9,25)(10,21)(11,22)(12,27)(14,18)(15,32)(16,33)(17,31)(20,26)(23,36)(24,35)(28,30)(29,34);(5,8,11,10)(6,14,17,16)(7,19,22,21)(9,23,26,24)(12,28,29,15)(13,18,31,33)(20,35,25,36)(27,30,34,32)Polycyclic presentation:
2^9|d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=di,e^b=eh,f^b=fi,h^b=hi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249861, #72249868, #72249884, … |
| maximal normal subgroup of | #72249861, #72249868, #72249884, … |
| maximal quotient of | #72265631, C2×#498833 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406029);