Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
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 | |
1712 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)(7,19)(8,23)(9,15)(10,18)(11,17)(12,16)(13,21)(14,27)(20,31)(22,30)(24,29)(25,32)(26,33)(28,34)(35,36);(5,7)(6,13)(8,24)(9,10)(11,22)(12,21)(14,31)(15,17)(16,30)(18,19)(20,26)(23,36)(25,27)(28,35)(29,33)(32,34);(2,4)(5,8)(6,14)(7,20)(9,25)(10,26)(11,27)(12,28)(13,29)(15,32)(16,33)(17,23)(18,34)(19,31)(21,24)(22,35)(30,36);(5,9,12,22,11,21,10,7)(6,15,18,19,17,13,16,30)(8,25,28,35,27,24,26,20)(14,32,34,31,23,29,33,36)Polycyclic presentation:
2^9|d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,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, 325681);