Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417541 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2408 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,25,11,14,28,13)(7,15,17,9,18,27,32,24)(8,19,22,30,23,12,26,31)(16,29,20,21,34,33,35,36);(1,2)(3,4)(5,7,10,17,11,18,28,32)(6,9,13,15,14,24,25,27)(8,20,22,34,23,35,26,16)(12,29,30,36,19,33,31,21);(2,4)(5,8)(6,12)(7,16)(9,21)(10,26)(11,23)(13,31)(14,19)(15,33)(17,35)(18,34)(20,32)(22,28)(24,36)(25,30)(27,29);(5,9)(6,7)(8,21)(10,27)(11,24)(12,16)(13,32)(14,18)(15,28)(17,25)(19,34)(20,31)(22,33)(23,36)(26,29)(30,35)Polycyclic presentation:
2^9|a^2=ef,b^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250092, #72252977, #72256869 |
| maximal normal subgroup of | #72250092, #72252977, #72256869 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326394);