Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454369 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,15)(8,12)(9,16)(10,18)(11,14)(13,20)(17,19);(1,3,4,2)(5,7,15,6,11,18,10,14)(8,13,20,12,19,16,9,17);(2,3)(5,8)(6,12)(7,16)(9,15)(10,20)(11,19)(13,18)(14,17);(5,9,15,8,11,20,10,19)(6,13,18,12,14,16,7,17)Polycyclic presentation:
2^9|a^2=f,b^2=efi,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338518, #72338520, #72338522 |
| maximal normal subgroup of | #72338518, #72338520, #72338522 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364429);