Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,16)(12,17)(13,19)(14,20)(15,18)(21,23)(22,24);(2,4)(9,11)(10,16)(12,13)(14,15)(17,19)(18,20);(6,8)(9,12,15,13)(10,17,18,19)(11,21,14,22)(16,23,20,24);(10,18)(11,14)(12,21)(13,22)(17,24)(19,23);(9,13,15,12)(10,19,18,17)(11,22,14,21)(16,24,20,23)Polycyclic presentation:
2^9|c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72282583, #72282638, #72282700, … |
| maximal normal subgroup of | #72282583, #72282638, #72282700, … |
| maximal quotient of | #72306408, #72306466, #72307726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480702);