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 | |
1356 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)(7,17)(8,21)(9,14)(10,16)(11,18)(12,20)(13,23)(15,26)(19,31)(22,32)(24,33)(25,28)(27,29)(30,34)(35,36);(1,2)(3,4)(5,7)(6,12)(8,19)(10,11)(13,28)(14,17)(15,18)(16,29)(20,26)(21,30)(23,24)(31,32)(33,36)(34,35);(2,4)(5,8)(6,13)(7,19)(9,25)(10,23)(11,24)(12,28)(14,30)(15,32)(16,33)(17,21)(18,31)(20,35)(22,27)(26,34)(29,36);(6,14)(7,10)(9,26)(12,16)(13,30)(15,17)(18,29)(19,23)(20,27)(21,32)(22,35)(25,34)(28,33)(31,36)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,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 | #72249915 |
| maximal normal subgroup of | #72249915 |
| maximal quotient of | #72308622 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406959);