Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1176 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)(7,11)(8,12,19,27)(9,20,21,13)(10,14)(15,22,29,25)(16,30,31,23)(17,24)(18,32)(26,34)(28,36)(33,35);(1,2)(3,4)(5,7,10,17)(6,11,14,24)(8,15,19,29)(9,16,21,31)(12,25,27,22)(13,23,20,30)(18,33,34,28)(26,35,32,36);(2,4)(5,8)(6,12,14,27)(7,15)(9,18)(10,19)(11,22,24,25)(13,26,20,32)(16,28)(17,29)(21,34)(23,35,30,36)(31,33);(5,9)(6,13)(7,16)(8,18)(10,21)(11,23)(12,26)(14,20)(15,28)(17,31)(19,34)(22,35)(24,30)(25,36)(27,32)(29,33)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ef,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,h^c=hi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283596, #72283700, #72283817, … |
| maximal normal subgroup of | #72283596, #72283700, #72283817, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291446);