Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,16,12,18,10,17)(7,19,22,30,9,15,21,13)(8,23,26,14,28,32,27,33)(20,35,24,34,25,29,36,31);(5,7,10,21,12,9,11,22)(6,13,16,19,18,30,17,15)(8,24,26,25,28,36,27,20)(14,31,23,29,33,34,32,35);(1,3,4,2)(5,8)(6,14)(7,20)(9,25)(10,26)(11,27)(12,28)(13,29)(15,31)(16,23)(17,32)(18,33)(19,34)(21,24)(22,36)(30,35);(5,9)(6,15)(7,12)(8,25)(10,21)(11,22)(13,17)(14,31)(16,30)(18,19)(20,28)(23,35)(24,26)(27,36)(29,32)(33,34)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296323);