Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| socle | C22 | C2×#393 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,18,21,12)(9,13)(10,14)(15,28,19,23)(16,24)(17,25)(20,31,33,26)(22,27)(29,36,32,34)(30,35);(1,2,3,4)(5,7)(6,11)(8,19,21,15)(9,16)(10,17)(12,23,18,28)(13,24)(14,25)(20,32,33,29)(22,30)(26,34,31,36)(27,35);(5,8)(6,12)(7,15)(9,20)(10,21)(11,23)(13,26)(14,18)(16,29)(17,19)(22,33)(24,34)(25,28)(27,31)(30,32)(35,36);(5,9,10,22)(6,13,14,27)(7,16,17,30)(8,20,21,33)(11,24,25,35)(12,26,18,31)(15,29,19,32)(23,34,28,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72303853, #72311760, #72311770, … |
| maximal normal subgroup of | #72303853, #72311760, #72311770, … |
| maximal quotient of | #72312178, #72312211, #72312650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264962);