Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499757 |
| 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 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1420 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)(8,12)(9,16)(10,15)(11,17)(13,20)(14,22)(19,32)(21,33)(23,30)(24,28)(25,29)(26,31)(27,36)(34,35);(1,2)(3,4)(6,12)(8,22)(9,10)(13,27)(14,16)(15,28)(17,29)(18,32)(19,35)(20,26)(23,36)(24,25)(30,33)(31,34);(2,4)(5,7)(6,13)(8,23)(9,26)(10,20)(11,21)(12,27)(14,32)(15,33)(16,18)(17,31)(19,25)(22,36)(24,35)(28,30)(29,34);(6,12)(8,22)(9,10)(13,27)(14,16)(15,28)(17,29)(18,32)(19,35)(20,26)(23,36)(24,25)(30,33)(31,34)Polycyclic presentation:
2^9|e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,e^b=eh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,e^d=eh,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249915, #72250402, #72319933, … |
| maximal normal subgroup of | #72249915, #72250402, #72319933, … |
| maximal quotient of | #72308613, C2×#499757 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406953);