Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421163 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1332 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)(7,17,21,26)(8,22)(9,12)(10,15)(11,16)(13,20,29,18)(14,23)(19,33,34,28)(24,30)(25,27)(31,35,36,32);(6,12)(7,18)(8,23)(9,10)(15,16)(17,26)(19,34)(20,21)(22,27)(24,25)(28,31)(33,36);(5,7)(6,13)(8,19)(9,18)(10,20)(11,21)(12,26)(14,28)(15,17)(16,29)(22,31)(23,32)(24,35)(25,34)(27,36)(30,33);(2,4)(5,8,10,24,11,25,9,23)(6,14,12,27,16,30,15,22)(7,19,20,35,21,34,18,32)(13,28,26,36,29,33,17,31)Polycyclic presentation:
2^9|a^2=gh,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358246);