Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,14,11,15,9,25)(7,16,18,12,20,27,19,28)(8,21,24,29,22,35,23,13)(17,26,32,34,31,36,33,30);(1,2,3,4)(5,7)(6,12)(8,17)(9,19)(10,18)(11,20)(13,26)(14,27)(15,28)(16,25)(21,34)(22,31)(23,33)(24,32)(29,36)(30,35);(5,8)(6,13,15,29)(7,17)(9,23)(10,24)(11,22)(12,26,28,36)(14,21,25,35)(16,30,27,34)(18,32)(19,33)(20,31);(8,22)(13,29)(17,31)(21,35)(23,24)(26,36)(30,34)(32,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72279346, #72279500, #72279945, … |
| maximal normal subgroup of | #72279346, #72279500, #72279945, … |
| maximal quotient of | #72280520, #72280683, #72280793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261312);