Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219508 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1258 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)(7,15,18,28)(8,12)(9,13)(10,14)(11,17,23,22)(16,29,31,35)(19,25)(20,26)(21,27)(24,30,34,32)(33,36);(5,7)(6,11)(8,16)(9,22)(10,18)(12,24)(13,28)(14,23)(15,26)(17,20)(19,32)(21,31)(25,35)(27,34)(29,36)(30,33);(2,4)(8,19)(9,20)(11,23)(12,25)(13,26)(16,30)(17,22)(21,33)(24,35)(27,36)(29,34)(31,32);(5,8,20,33,10,21,9,19)(6,12,26,36,14,27,13,25)(7,16,17,30,18,31,22,32)(11,24,15,29,23,34,28,35)Polycyclic presentation:
2^9|a^2=h,d^2=efhi,e^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72226670, #72226674, #72226676 |
| maximal normal subgroup of | #72226670, #72226674, #72226676 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289080);