Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417732 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,14)(9,12)(10,15)(11,16)(13,18)(19,33)(20,27)(21,25)(22,30)(23,26)(24,29)(28,35)(31,34)(32,36);(1,3,4,2)(6,12)(7,18)(8,22)(9,10)(13,27)(14,29)(15,16)(19,34)(20,21)(23,24)(28,33)(31,36);(2,3)(5,7)(6,13)(8,19)(9,18)(10,20)(11,21)(12,25)(14,28)(15,17)(16,27)(22,35)(23,32)(24,34)(26,33)(29,36)(30,31);(5,8,11,24)(6,14,16,29)(7,19,21,34)(9,23,10,22)(12,26,15,30)(13,28,27,36)(17,31,25,33)(18,32,20,35)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,d^2=i,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,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,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, 328970);