Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,15,9,14)(7,17,19,31,21,12,20,28)(8,22,24,32,25,13,26,30)(18,27,35,29,34,33,23,36);(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,29)(14,28)(15,17)(16,31)(18,26)(22,36)(24,34)(25,35)(27,30)(32,33);(2,4)(5,8)(6,13)(7,18)(9,24)(10,26)(11,25)(12,27)(14,30)(15,22)(16,32)(17,33)(19,23)(20,35)(21,34)(28,36)(29,31);(5,9,11,10)(6,14,15,16)(7,19,21,20)(8,24,25,26)(12,28,17,31)(13,30,22,32)(18,23,34,35)(27,36,33,29)Polycyclic presentation:
2^9|a^2=h,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250584, #72250603, #72250614, … |
| maximal normal subgroup of | #72250584, #72250603, #72250614, … |
| maximal quotient of | #72267263, C2×#357591 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264787);