Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454039 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2150 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)(5,6,12,8)(7,17,21,29)(9,24,26,30)(10,16,11,15)(13,22,18,20)(14,25,23,27)(19,32,34,28)(31,33,36,35);(5,7)(6,13)(8,18)(9,19)(10,22)(11,20)(12,21)(14,28)(15,29)(16,17)(23,32)(24,36)(25,35)(26,34)(27,33)(30,31);(5,8,12,6)(7,18,21,13)(9,24,26,30)(10,15,11,16)(14,25,23,27)(17,22,29,20)(19,31,34,36)(28,33,32,35);(2,4)(5,9)(6,14)(7,19)(8,23)(10,27)(11,25)(12,26)(13,28)(15,24)(16,30)(17,31)(18,32)(20,35)(21,34)(22,33)(29,36)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72228038 |
| maximal normal subgroup of | #72228038 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361288);