Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#397 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1622 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)(8,12)(9,22)(10,14)(11,19)(13,16)(17,30)(18,31)(20,29)(21,28)(23,27)(24,35)(25,32)(26,34)(33,36);(5,7,16,32,10,18,9,19)(6,11,22,31,14,25,13,15)(8,17,20,34,21,33,23,35)(12,26,27,30,28,24,29,36);(5,8,16,20,10,21,9,23)(6,12,22,27,14,28,13,29)(7,17,32,34,18,33,19,35)(11,24,31,36,25,26,15,30);(7,18)(11,25)(15,31)(17,33)(19,32)(24,26)(30,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=ef,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^c=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250584, #72250589, #72250605, … |
| maximal normal subgroup of | #72250584, #72250589, #72250605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265804);