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 | #1445 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,17,20,12)(8,21)(9,16)(10,15)(11,14)(13,23)(18,34,35,29)(19,28,26,32)(22,33,36,27)(24,30)(25,31);(5,7)(6,12)(8,22,25,36)(9,26)(10,19)(11,20)(13,29,30,34)(14,17)(15,32)(16,28)(18,24,35,23)(21,27,31,33);(1,2,3,4)(5,8,9,23,11,25,10,24)(6,13,15,21,14,30,16,31)(7,18,19,22,20,35,26,36)(12,27,28,29,17,33,32,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,e^b=ei,g^b=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237652, #72237658, #72237662, … |
| maximal normal subgroup of | #72237652, #72237658, #72237662, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31122);