Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417661 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,15,18,11)(8,12,20,30)(9,22)(10,13)(14,25)(16,26)(17,27)(19,28,23,32)(21,33,24,31)(29,36)(34,35);(5,7)(6,11)(8,17)(9,23)(10,18)(12,29)(13,15)(14,32)(16,21)(19,25)(20,36)(22,28)(24,35)(26,31)(27,30)(33,34);(2,4)(5,8)(6,12,13,30)(7,16)(9,24)(10,20)(11,26,15,34)(14,33,22,31)(17,23)(18,35)(19,36)(21,25)(27,32,29,28);(5,8)(6,12,13,30)(7,17)(9,21)(10,20)(11,27,15,29)(14,31,22,33)(16,23)(18,36)(19,35)(24,25)(26,32,34,28)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=e,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,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327722);