Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4438 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)(7,16)(8,20,23,13)(9,14)(10,24)(11,15)(12,18)(17,31,33,30)(19,27)(21,35,22,29)(25,28)(26,32,34,36);(1,2)(3,4)(5,7)(6,12)(8,17)(9,18)(10,25)(11,19)(13,26)(14,27)(15,28)(16,24)(20,34)(21,32)(22,36)(23,33)(29,31)(30,35);(2,4)(5,8)(6,13)(7,17)(9,21)(10,22)(11,23)(12,26)(14,29)(15,20)(16,30)(18,32)(19,33)(24,35)(25,36)(27,31)(28,34);(5,9,11,10)(6,14,15,24)(7,18,19,25)(8,21,23,22)(12,27,28,16)(13,29,20,35)(17,32,33,36)(26,31,34,30)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237508, #72237512, #72250089, … |
| maximal normal subgroup of | #72237508, #72237512, #72250089, … |
| maximal quotient of | #72250431, #72257703, #72257783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264666);