Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455641 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
895 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,12,18,11,17)(7,19,22,33,24,26,23,13)(8,25,27,14,29,34,28,21)(9,15,30,20,31,36,32,35);(1,2)(3,4)(5,7,11,22,12,24,10,23)(6,13,17,19,18,33,16,26)(8,15,28,20,29,36,27,35)(9,21,30,34,31,14,32,25);(2,4)(5,8,12,29)(6,14,18,21)(7,20,24,35)(9,13,31,33)(10,28,11,27)(15,23,36,22)(16,25,17,34)(19,30,26,32);(5,9)(6,15)(7,21)(8,26)(10,32)(11,30)(12,31)(13,28)(14,24)(16,35)(17,20)(18,36)(19,29)(22,34)(23,25)(27,33)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362838);