Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458125 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2495 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,11,10,15,25,14)(7,16,17,26,19,12,24,32)(8,20,22,31,23,13,29,27)(18,33,21,30,34,28,36,35);(1,2,3,4)(6,11,15,14)(7,17,19,24)(12,16)(13,27,20,31)(18,21,34,36)(26,32)(28,33)(30,35);(2,4)(5,7)(6,12)(8,21)(9,24)(10,19)(11,26)(13,30)(14,32)(15,16)(17,25)(18,22)(20,35)(23,36)(27,33)(28,31)(29,34);(5,8,9,22,10,23,25,29)(6,13,11,27,15,20,14,31)(7,18,17,21,19,34,24,36)(12,28,32,35,16,33,26,30)Polycyclic presentation:
2^9|a^2=ef,b^2=efgh,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380691);