Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196278 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3842 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)(8,18,20,30)(9,13)(10,14)(12,19,28,22)(15,32,34,36)(16,25)(17,26)(21,29)(23,31)(24,33,27,35);(2,4)(5,7,10,17)(6,11,14,26)(8,15,20,34)(9,21,23,16)(12,27,28,24)(13,29,31,25)(18,32,30,36)(19,35,22,33);(5,8)(6,12)(7,15)(9,22)(10,20)(11,24)(13,30)(14,28)(16,35)(17,34)(18,31)(19,23)(21,33)(25,36)(26,27)(29,32);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72302333, #72302890, #72302893 |
| maximal normal subgroup of | #72302333, #72302890, #72302893 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266707);