Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#393 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1403 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)(7,12)(8,20,22,31)(9,14)(10,25)(11,15)(13,23,30,24)(16,29,32,36)(17,27)(18,28)(19,34)(21,26,33,35);(5,7)(6,12)(8,21,22,33)(9,18)(10,19)(11,17)(13,29,30,36)(14,28)(15,27)(16,24,32,23)(20,35,31,26)(25,34);(5,8,9,24,11,22,10,23)(6,13,14,31,15,30,25,20)(7,16,18,21,17,32,19,33)(12,26,28,36,27,35,34,29);(2,4)(7,17)(12,27)(16,32)(18,19)(21,33)(26,35)(28,34)(29,36)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=di,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286161, #72286979, #72287097 |
| maximal normal subgroup of | #72286161, #72286979, #72287097 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305292);