Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1944 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)(5,6,10,16,12,17,11,30)(7,18,20,33,21,32,29,13)(8,22,24,14,19,35,25,31)(9,26,27,34,28,15,23,36);(5,7)(6,13)(8,23)(9,25)(10,29)(11,20)(12,21)(14,34)(15,22)(16,32)(17,33)(18,30)(19,27)(24,28)(26,35)(31,36);(5,8,12,19)(6,14,17,31)(7,9,21,28)(10,24,11,25)(13,15,33,26)(16,35,30,22)(18,36,32,34)(20,27,29,23);(5,9)(6,15)(7,19)(8,21)(10,23)(11,27)(12,28)(13,31)(14,33)(16,34)(17,26)(18,35)(20,24)(22,32)(25,29)(30,36)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265649, #72265655, #72265688, … |
| maximal normal subgroup of | #72265649, #72265655, #72265688, … |
| maximal quotient of | #72286809 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290264);