Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457102 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(8,22)(9,11)(13,17)(15,16)(18,34)(19,20)(23,35)(24,25)(26,32)(27,36)(28,29);(1,2,3,4)(5,6,11,16,10,15,9,12)(7,17,20,29,21,28,19,13)(8,14,22,32,25,31,24,26)(18,33,34,36,35,30,23,27);(2,4)(5,7,11,19,10,21,9,20)(6,13,16,28,15,29,12,17)(8,23,22,35,25,34,24,18)(14,30,32,36,31,33,26,27);(5,8,10,25)(6,14,15,31)(7,18)(9,24,11,22)(12,26,16,32)(13,27)(17,33)(19,23)(20,34)(21,35)(28,30)(29,36)Polycyclic presentation:
2^9|b^2=g,c^2=ef,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373951);