Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396113 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| 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 | |
995 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16,12,18,10,17)(7,19,22,14,8,23,21,13)(9,25,27,32,28,15,26,31)(20,34,36,29,24,33,35,30);(1,2)(3,4)(5,7,10,21,12,8,11,22)(6,13,16,19,18,14,17,23)(9,20,26,35,28,24,27,36)(15,29,31,33,25,30,32,34);(2,4)(5,8,12,7)(6,14,18,13)(9,24,28,20)(10,21,11,22)(15,30,25,29)(16,19,17,23)(26,35,27,36)(31,33,32,34);(5,9)(6,15,18,25)(7,20)(8,24)(10,26)(11,27)(12,28)(13,29,14,30)(16,31,17,32)(19,33,23,34)(21,35)(22,36)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307703);