Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6,8,10)(7,11,12,9)(13,14,17,16)(15,23,25,20)(18,22,19,27)(21,28,24,26);(2,3)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,23)(17,25)(18,28)(19,26)(21,22)(24,27);(1,2,4,3)(6,10)(9,11)(13,16)(14,17)(15,24)(18,22)(19,27)(20,26)(21,25)(23,28);(1,3,4,2)(7,12)(9,11)(13,15,17,25)(14,21,16,24)(18,26,19,28)(20,27,23,22)Polycyclic presentation:
2^9|a^2=e,b^2=fi,c^2=ef,d^2=efi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339157);