Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394046 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62853 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3503 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17,18,12)(8,20,21,13)(9,23)(10,26)(11,15)(14,25)(16,30)(19,33,27,34)(22,35,28,32)(24,31)(29,36);(1,2)(3,4)(5,7)(6,12)(8,9)(10,27)(11,18)(13,31)(14,32)(15,17)(16,34)(19,30)(20,23)(21,24)(22,29)(25,28)(26,33)(35,36);(2,4)(5,8)(6,13)(7,9)(10,28)(11,21)(12,31)(14,34)(15,20)(16,32)(17,23)(18,24)(19,29)(22,30)(25,27)(26,35)(33,36);(5,9)(6,14)(7,8)(10,29)(11,24)(12,32)(13,34)(15,36)(16,31)(17,35)(18,21)(19,28)(20,33)(22,27)(23,26)(25,30)Polycyclic presentation:
2^9|a^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282010 |
| maximal normal subgroup of | #72282010 |
| maximal quotient of | #72282252 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302647);