Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417738 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| 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 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,16)(7,17,21,26)(8,22,25,14)(9,12,10,15)(13,19,29,20)(18,31,34,36)(23,27,24,30)(28,33,35,32);(6,12)(8,23)(9,10)(13,26)(14,22)(15,16)(17,29)(18,32)(19,20)(24,25)(31,36)(33,34);(1,2,3,4)(5,7)(6,13)(8,18)(9,20)(10,19)(11,21)(12,26)(14,28)(15,17)(16,29)(22,35)(23,33)(24,32)(25,34)(27,36)(30,31);(5,8)(6,14)(7,18)(9,24)(10,23)(11,25)(12,27)(13,28)(15,30)(16,22)(17,31)(19,33)(20,32)(21,34)(26,36)(29,35)Polycyclic presentation:
2^9|a^2=i,c^2=ef,e^2=i,f^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,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330392);