Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23444 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4422 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)(5,6,9,14,10,13,22,28)(7,15,18,29,17,11,21,24)(8,12,20,27,19,26,32,36)(16,23,31,34,30,25,33,35);(2,4)(5,7)(6,11)(8,16)(9,21)(10,17)(12,25)(13,15)(14,29)(18,22)(19,30)(20,33)(23,26)(24,28)(27,34)(31,32)(35,36);(5,8)(6,12)(7,16)(9,20)(10,19)(11,23)(13,26)(14,27)(15,25)(17,30)(18,31)(21,33)(22,32)(24,34)(28,36)(29,35);(8,19)(12,26)(16,30)(20,32)(23,25)(27,36)(31,33)(34,35)Polycyclic presentation:
2^9|a^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238124, #72249697, #72249737, … |
| maximal normal subgroup of | #72238124, #72249697, #72249737, … |
| maximal quotient of | #72238127, #72280566, #72302538 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264730);