Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456658 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1087 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,18)(8,24)(9,28)(10,16)(11,15)(12,17)(13,21)(14,26)(19,35)(20,34)(22,31)(23,32)(25,29)(27,33)(30,36);(5,7)(6,13,17,18)(8,25)(9,20)(10,22)(11,23)(12,21)(14,30,33,35)(15,31,16,32)(19,26)(24,34,28,29)(27,36);(2,4)(5,8,12,9)(6,14,17,33)(7,19,21,36)(10,27,11,26)(13,29,18,34)(15,28,16,24)(20,22,25,23)(30,31,35,32);(5,9,12,8)(6,14,17,33)(7,20,21,25)(10,26,11,27)(13,30,18,35)(15,28,16,24)(19,23,36,22)(29,32,34,31)Polycyclic presentation:
2^9|b^2=egh,c^2=i,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378643);