Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2482 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1516 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)(7,11)(8,18,20,28)(9,13)(10,14)(12,21,27,22)(15,30,19,34)(16,25)(17,26)(23,29)(24,31,35,32)(33,36);(5,7,9,16,10,17,23,33)(6,11,13,25,14,26,29,36)(8,19,21,32,20,15,22,31)(12,24,18,30,27,35,28,34);(5,8)(6,12)(7,15)(9,22)(10,20)(11,24)(13,28)(14,27)(16,32)(17,19)(18,29)(21,23)(25,34)(26,35)(30,36)(31,33);(2,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,24)(16,25)(17,26)(18,21)(19,35)(20,27)(22,28)(23,29)(30,31)(32,34)(33,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321973 |
| maximal normal subgroup of | #72321973 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328096);