Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418508 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2017 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1623 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,18)(8,9)(10,25)(11,16)(12,17)(13,21)(14,27)(15,30)(19,36)(20,34)(22,33)(23,32)(24,35)(26,29)(28,31);(5,7)(6,13)(8,24)(9,28)(10,22)(11,21)(12,23)(14,20)(15,34)(16,32)(17,33)(18,25)(19,26)(27,31)(29,35)(30,36);(2,4)(5,8)(6,14)(7,19)(9,25)(10,26)(11,27)(12,15)(13,28)(16,30)(17,29)(18,35)(20,32)(21,24)(22,34)(23,31)(33,36);(5,9,10,14,12,30,11,29)(6,15,16,27,17,8,25,26)(7,20,21,28,23,35,22,36)(13,31,18,34,33,19,32,24)Polycyclic presentation:
2^9|d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325704);