Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1393 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)(7,17)(8,23,26,28)(9,25,29,14)(10,24)(11,15)(12,16)(13,20)(18,34,35,27)(19,31,36,33)(21,30)(22,32);(2,4)(5,7,12,22)(6,13,16,30)(8,18,26,35)(9,27,29,34)(10,21,11,20)(14,31,25,33)(15,32,24,17)(19,23,36,28);(1,2,3,4)(5,8,11,25,12,26,10,14)(6,9,24,28,16,29,15,23)(7,18,21,31,22,35,20,33)(13,27,32,36,30,34,17,19);(5,9)(6,14)(7,19)(8,24)(10,23)(11,28)(12,29)(13,18)(15,26)(16,25)(17,33)(20,34)(21,27)(22,36)(30,35)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265644, #72265649, #72265666 |
| maximal normal subgroup of | #72265644, #72265649, #72265666 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293106);