Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1588 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)(5,6,12,16,11,17,10,30)(7,18,21,33,22,32,20,13)(8,23,26,14,27,31,25,34)(9,28,19,35,29,36,24,15);(1,2,4,3)(5,7,10,20,11,22,12,21)(6,13,16,18,17,33,30,32)(8,24,25,29,27,19,26,9)(14,15,31,28,34,35,23,36);(5,8)(6,14)(7,19)(9,20)(10,25)(11,27)(12,26)(13,15)(16,31)(17,34)(18,28)(21,29)(22,24)(23,30)(32,36)(33,35);(1,3,4,2)(5,9,10,24,11,29,12,19)(6,15,16,28,17,35,30,36)(7,8,20,25,22,27,21,26)(13,31,18,34,33,23,32,14)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72279578, #72279626, #72280138, … |
| maximal normal subgroup of | #72279578, #72279626, #72280138, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267414);