Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2053 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)(7,17)(8,22)(9,15)(10,14)(11,16)(12,20)(13,23)(18,29)(19,28)(21,27)(24,31)(25,30)(26,35)(32,34)(33,36);(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,29)(14,27)(15,17)(16,28)(22,32)(23,33)(24,35)(25,34)(26,30)(31,36);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,22)(15,31)(16,30)(17,32)(19,33)(20,34)(21,35)(27,36)(28,29);(7,19)(9,10)(12,27)(14,15)(17,28)(18,33)(20,21)(22,31)(23,25)(26,36)(29,32)(34,35)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72255707, #72256284, #72256291 |
| maximal normal subgroup of | #72255707, #72256284, #72256291 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290360);