Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #387 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1442 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,15,18,11)(8,19)(9,23)(10,13)(12,21)(14,24)(16,30,20,34)(17,26,32,29)(22,33)(25,31,27,35)(28,36);(1,2)(3,4)(5,7)(6,11)(8,20,22,16)(9,17)(10,18)(12,27,28,25)(13,15)(14,26)(19,30,33,34)(21,35,36,31)(23,29)(24,32);(2,4)(5,8)(6,12,13,28)(7,16)(9,21)(10,22)(11,25,15,27)(14,19,23,33)(17,31)(18,20)(24,36)(26,34,29,30)(32,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,f^b=fi,g^b=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265531, #72265544, #72265573, … |
| maximal normal subgroup of | #72265531, #72265544, #72265573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30964);