Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394034 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3469 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)(5,6,9,24,11,13,10,15)(7,16,20,27,19,30,17,12)(8,21,22,28,23,14,25,29)(18,31,32,26,34,36,33,35);(5,7)(6,12)(8,18)(9,20)(10,17)(11,19)(13,27)(14,26)(15,30)(16,24)(21,35)(22,32)(23,34)(25,33)(28,31)(29,36);(2,4)(6,13)(7,17)(9,10)(12,16)(14,21)(18,32)(19,20)(22,25)(26,31)(27,30)(33,34)(35,36);(5,8)(6,14)(7,18)(9,25)(10,22)(11,23)(12,26)(13,21)(15,29)(16,31)(17,32)(19,34)(20,33)(24,28)(27,35)(30,36)Polycyclic presentation:
2^9|a^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302626);