Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188764 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#383 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1262 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)(7,16)(8,20,22,27)(9,14)(10,15)(11,19)(12,21,28,34)(13,23)(17,24,30,35)(18,26)(25,32)(29,31,33,36);(5,7,10,18)(6,11)(8,17,22,30)(9,19,23,32)(12,24)(13,16)(14,26)(15,25)(20,33)(21,31,34,36)(27,29)(28,35);(5,8,9,21,10,22,23,34)(6,12,13,27,15,28,14,20)(7,17,19,31,18,30,32,36)(11,24,16,29,25,35,26,33);(2,4)(7,18)(11,25)(16,26)(17,30)(19,32)(24,35)(29,33)(31,36)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,d^b=di,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224742, #72224750, #72224752, … |
| maximal normal subgroup of | #72224742, #72224750, #72224752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299544);