Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,16,12,17,10,18)(7,14,21,29,8,22,20,13)(9,15,26,32,27,33,25,34)(19,30,35,28,23,31,24,36);(1,2,3,4)(5,7,10,20,12,8,11,21)(6,13,16,14,17,29,18,22)(9,24,25,23,27,35,26,19)(15,31,32,36,33,30,34,28);(5,8,12,7)(6,14,17,22)(9,23,27,19)(10,20,11,21)(13,18,29,16)(15,30,33,31)(24,26,35,25)(28,32,36,34);(2,4)(5,9)(6,15,17,33)(7,19)(8,23)(10,26)(11,25)(12,27)(13,28,29,36)(14,30,22,31)(16,34,18,32)(20,35)(21,24)Polycyclic presentation:
2^9|a^2=fg,b^2=e,c^2=i,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343358);