Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | #23406 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1430 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)(7,17)(8,22,26,31)(9,14)(10,15)(11,16)(12,20)(13,24,32,25)(18,27,33,36)(19,28)(21,29)(23,30,34,35);(2,4)(5,7)(6,12,16,29)(8,23,26,34)(9,20)(10,21)(11,19)(13,30)(14,28,15,17)(18,25,33,24)(22,27)(31,36)(32,35);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,22)(15,31)(16,32)(17,30)(19,33)(20,34)(21,23)(28,35)(29,36);(2,4)(6,14,16,15)(7,19)(12,17,29,28)(13,22,32,31)(18,33)(20,21)(23,34)(27,30,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=efh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280601, #72318910, #72318940, … |
| maximal normal subgroup of | #72280601, #72318910, #72318940, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285302);