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×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1278 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)(7,19)(8,23,27,33)(9,13)(10,17)(11,16)(12,18)(14,26,32,25)(15,22)(20,34,36,28)(21,29)(24,30,35,31);(5,7,12,22)(6,13,18,29)(8,24,27,35)(9,10,21,11)(14,30,32,31)(15,17,19,16)(20,26,36,25)(23,28,33,34);(5,8)(6,14)(7,20)(9,24)(10,25)(11,26)(12,27)(13,28)(15,31)(16,23)(17,33)(18,32)(19,30)(21,35)(22,36)(29,34);(2,4)(5,9,10,22,12,21,11,7)(6,15,16,13,18,19,17,29)(8,24,25,36,27,35,26,20)(14,31,23,28,32,30,33,34)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=e,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, 358247);