Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,17)(7,18)(8,25,28,21)(9,20)(10,15,19,16)(12,23)(13,26,33,27)(14,30)(22,32)(24,35)(29,34)(31,36);(1,2,3,4)(5,7,19,22,11,24,10,23)(6,12,16,18,17,32,15,35)(8,20,27,34,28,36,26,14)(9,21,29,33,31,25,30,13);(5,8)(6,13)(7,20)(9,12)(10,26)(11,28)(14,23)(15,25)(16,21)(17,33)(18,29)(19,27)(22,34)(24,36)(30,35)(31,32);(5,9)(6,14)(7,21)(8,12)(10,30)(11,31)(13,23)(15,36)(16,20)(17,34)(18,27)(19,29)(22,33)(24,25)(26,35)(28,32)Polycyclic presentation:
2^9|a^2=h,b^2=eh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,e^b=ei,g^b=gi,h^b=hi,f^c=fi,e^d=ei,g^d=gi,h^d=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72250134, #72257145, #72257146, … |
| maximal normal subgroup of | #72250134, #72257145, #72257146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299849);