Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499763 |
| 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 | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,17)(8,20)(9,10)(13,29)(14,21)(15,27)(16,28)(18,22)(19,25)(23,24)(26,31)(30,33)(32,35);(1,2)(3,4)(5,6,11,16)(7,13,19,32)(8,21,24,27)(9,15,10,14)(12,23,28,20)(17,33,25,31)(18,35,22,29)(26,34,30,36);(2,4)(5,7)(6,13)(8,22)(9,25)(10,17)(11,19)(12,26)(14,33)(15,31)(16,32)(18,24)(20,36)(21,29)(23,34)(27,35)(28,30);(5,6)(7,13)(8,21)(9,15)(10,14)(11,16)(12,20)(17,33)(18,35)(19,32)(22,29)(23,28)(24,27)(25,31)(26,36)(30,34)Polycyclic presentation:
2^9|b^2=i,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,d^b=di,e^b=eh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,e^d=eh,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72250402 |
| maximal normal subgroup of | #72250402 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406960);