Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,17,19,28)(8,22,25,30)(9,15)(10,16)(11,14)(12,20,27,21)(13,24,29,23)(18,32)(26,33)(31,34)(35,36);(1,2,3,4)(5,7,9,21,11,19,10,20)(6,12,15,28,14,27,16,17)(8,18,24,34,25,33,23,35)(13,26,22,36,29,32,30,31);(5,8)(6,13)(7,18)(9,24)(10,23)(11,25)(12,26)(14,29)(15,22)(16,30)(17,31)(19,33)(20,35)(21,34)(27,32)(28,36);(7,19)(12,27)(17,28)(18,33)(20,21)(26,32)(31,36)(34,35);(6,14)(12,27)(13,29)(15,16)(17,28)(22,30)(26,32)(31,36)Polycyclic presentation:
2^9|a^2=h,b^2=fh,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=di,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72237505, #72237506, #72237507, … |
| maximal normal subgroup of | #72237505, #72237506, #72237507, … |
| maximal quotient of | #72310292, #72310318, C2×#6371397, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278593);