Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417738 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2438 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)(8,14)(9,12)(10,15)(11,16)(13,20)(18,28)(19,33)(21,27)(22,30)(23,29)(24,25)(26,32)(31,35)(34,36);(1,2)(3,4)(6,12)(7,18)(8,22)(9,10)(14,29)(15,16)(17,27)(20,21)(23,24)(26,31)(32,34)(33,36);(5,7)(6,13)(8,19)(9,18)(10,20)(11,21)(12,17)(14,26)(15,27)(16,28)(22,34)(23,35)(24,32)(25,31)(29,36)(30,33);(2,4)(5,8)(6,14)(7,19)(9,24)(10,22)(11,23)(12,25)(13,26)(15,30)(16,29)(17,31)(18,32)(20,34)(21,35)(27,33)(28,36)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^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,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 | #72255770, #72256282, #72256291 |
| maximal normal subgroup of | #72255770, #72256282, #72256291 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324934);