Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1218 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)(7,17,20,25)(8,14)(9,15)(10,16)(11,13)(12,19,26,18)(21,31,34,35)(22,29)(23,28)(24,30)(27,33,36,32);(2,4)(5,7,9,21,10,19,24,33,11,20,23,34,8,18,22,32)(6,12,15,27,16,17,30,31,13,26,28,36,14,25,29,35);(7,18,20,19)(9,23)(12,25,26,17)(15,28)(21,33,34,32)(22,24)(27,31,36,35)(29,30);(6,13)(12,26)(14,16)(15,28)(17,25)(27,36)(29,30)(31,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=i,g^2=h,h^2=i,b^a=be,d^a=di,c^b=ch,e^b=ei,f^b=fi,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72231495, #72231501, #72231520, … |
| maximal normal subgroup of | #72231495, #72231501, #72231520, … |
| maximal quotient of | #72239329, #72280553, #72285281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400151);