Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417541 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,14,9,15,10,22)(7,12,23,26,17,27,18,30)(8,19,24,29,20,34,21,13)(16,31,35,28,32,36,33,25);(1,2)(3,4)(5,7)(6,12)(8,16)(9,17)(10,23)(11,18)(13,28)(14,30)(15,27)(19,31)(20,32)(21,35)(22,26)(24,33)(25,29)(34,36);(2,4)(5,8,9,20)(6,13)(7,16,17,32)(10,21,11,24)(12,25)(14,19)(15,29)(18,33,23,35)(22,34)(26,31)(27,28)(30,36);(7,17)(8,20)(13,29)(18,23)(19,34)(21,24)(25,28)(31,36)Polycyclic presentation:
2^9|a^2=ef,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285507, #72285552, #72285668, … |
| maximal normal subgroup of | #72285507, #72285552, #72285668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326640);