Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137661 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#3060 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,16,18,12)(8,20)(9,25)(10,14)(11,17,29,22)(13,23)(15,27)(19,32,26,34)(21,31,36,30)(24,28)(33,35);(5,7)(6,11)(8,21)(9,26)(10,18)(12,20)(13,32)(14,29)(15,31)(16,28)(17,35)(19,27)(22,23)(24,36)(25,30)(33,34);(2,4)(5,7)(6,12)(8,22)(9,19)(10,18)(11,28)(13,30)(14,16)(15,32)(17,24)(20,29)(21,23)(25,34)(26,27)(31,33)(35,36)Polycyclic presentation:
2^9|a^2=g,d^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,g^b=gi,h^b=hi,d^c=di,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231480, #72231524, #72231553, … |
| maximal normal subgroup of | #72231480, #72231524, #72231553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54835);