Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#576 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3555 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)(5,6,9,17,11,15,10,16)(7,18,21,30,23,12,22,31)(8,24,27,14,25,33,26,13)(19,32,36,28,35,29,20,34);(2,4)(5,7)(6,12)(8,20)(9,22)(10,21)(11,23)(13,32)(14,29)(15,18)(16,31)(17,30)(19,26)(24,34)(25,36)(27,35)(28,33);(5,8)(6,13)(7,19)(9,27)(10,26)(11,25)(12,28)(14,15)(16,33)(17,24)(18,34)(20,22)(21,36)(23,35)(29,31)(30,32);(5,8)(6,14)(7,20)(9,26)(10,27)(11,25)(12,29)(13,15)(16,33)(17,24)(18,32)(19,22)(21,35)(23,36)(28,31)(30,34)Polycyclic presentation:
2^9|a^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302609);