Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,16,9,25)(8,20,24,32)(10,14)(11,15)(12,18,29,19)(13,23,31,26)(17,28)(21,30)(22,35)(27,33)(34,36);(1,2,4,3)(5,7,11,9)(6,12,15,29)(8,21)(10,19,27,18)(13,30)(14,25,33,16)(17,26)(20,35)(22,23)(24,34)(28,32)(31,36);(5,8)(6,13)(7,17)(9,22)(10,26)(11,24)(12,28)(14,32)(15,31)(16,30)(18,21)(19,34)(20,33)(23,27)(25,36)(29,35);(1,3,4,2)(5,9,11,7)(6,12,15,29)(8,22,24,17)(10,18,27,19)(13,28,31,35)(14,25,33,16)(20,30,32,36)(21,23,34,26)Polycyclic presentation:
2^9|a^2=h,b^2=egh,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280644, #72280718 |
| maximal normal subgroup of | #72280644, #72280718 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275096);