Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2408 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2324 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)(7,17)(8,22,25,13)(9,15)(10,14)(11,16)(12,19)(18,30,34,33)(20,29)(21,28)(23,27,35,36)(24,31,26,32);(1,2,3,4)(5,7,9,19,11,21,10,20)(6,12,14,17,16,29,15,28)(8,23,24,34,25,35,26,18)(13,30,31,36,22,33,32,27);(5,6)(7,17)(8,13)(9,15)(10,14)(11,16)(12,19)(18,33)(20,29)(21,28)(22,25)(23,36)(24,32)(26,31)(27,35)(30,34);(2,4)(5,8)(6,13)(7,18)(9,26)(10,24)(11,25)(12,27)(14,32)(15,31)(16,22)(17,33)(19,35)(20,23)(21,34)(28,30)(29,36)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72323003 |
| maximal normal subgroup of | #72323003 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326228);