Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1435 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)(7,17,21,25)(8,14,23,29)(9,12)(10,15)(11,16)(13,20,28,18)(19,33)(22,30,24,26)(27,34)(31,35)(32,36);(2,4)(6,12)(7,18)(8,22)(9,10)(14,29)(15,16)(17,25)(20,21)(23,24)(27,33)(31,36)(32,34);(1,2,3,4)(5,7)(6,13)(8,19)(9,18)(10,20)(11,21)(12,25)(14,27)(15,17)(16,28)(22,34)(23,35)(24,32)(26,33)(29,36)(30,31);(5,8)(6,14)(7,19)(9,24)(10,22)(11,23)(12,26)(13,27)(15,30)(16,29)(17,31)(18,32)(20,34)(21,35)(25,33)(28,36)Polycyclic presentation:
2^9|a^2=h,c^2=egh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 297619);