Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417802 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2727 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,17)(8,22)(9,15)(10,16)(11,19)(12,25)(13,21)(14,27)(18,28)(20,29)(23,30)(24,32)(26,31)(33,34)(35,36);(1,2,3,4)(5,7,9,19,10,21,27,20)(6,11,14,17,16,29,15,13)(8,23,24,35,26,34,25,18)(12,30,31,28,32,33,22,36);(5,8)(6,12)(7,18)(9,24)(10,26)(11,28)(13,30)(14,31)(15,22)(16,32)(17,33)(19,23)(20,34)(21,35)(25,27)(29,36);(2,4)(5,7)(6,13)(8,18)(9,20)(10,21)(11,15)(12,30)(14,29)(16,17)(19,27)(22,28)(23,25)(24,34)(26,35)(31,36)(32,33)Polycyclic presentation:
2^9|b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72255739, #72256181, #72256189, … |
| maximal normal subgroup of | #72255739, #72256181, #72256189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324998);