Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420420 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,26,12,17,11,16)(7,15,9,25,20,33,19,13)(8,14,27,30,24,32,23,31)(18,34,21,36,35,29,22,28);(1,2)(3,4)(5,7,12,20)(6,13,17,25)(8,21,24,22)(9,11,19,10)(14,29,32,34)(15,16,33,26)(18,27,35,23)(28,31,36,30);(2,4)(5,8,10,23,12,24,11,27)(6,14,16,30,17,32,26,31)(7,18,19,21,20,35,9,22)(13,28,15,29,25,36,33,34);(5,9,10,7,12,19,11,20)(6,15,16,25,17,33,26,13)(8,22,23,18,24,21,27,35)(14,29,30,36,32,34,31,28)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=e,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327617);