Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1239 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)(7,15)(8,20)(9,25)(10,12)(11,18)(13,24)(14,26)(16,28)(17,27)(19,32)(21,33)(22,29)(23,31)(30,34)(35,36);(5,7)(6,11)(8,21)(9,16)(10,19)(12,28)(13,30)(14,32)(15,25)(17,22)(18,26)(20,36)(23,34)(24,35)(27,31)(29,33);(2,4)(6,12)(7,16,19,18)(11,15,28,32)(13,29)(14,25)(17,21,35,34)(20,31)(27,33,36,30);(2,4)(5,8)(6,13)(7,17)(9,24)(10,23)(11,27)(12,29)(14,20)(15,33)(16,21)(18,34)(19,35)(22,26)(25,31)(28,36)(30,32)Polycyclic presentation:
2^9|a^2=h,c^2=fhi,e^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,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296224);