Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2847 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(11,15)(14,16)(17,18)(19,21)(20,23)(22,24);(5,6,8,7)(10,13)(14,16)(18,19)(22,23);(1,2)(3,4)(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22);(17,20)(18,22)(19,23)(21,24);(9,11)(10,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|b^2=fi,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
| maximal subgroup of | #72326260, #72326261, #72326270, … |
| maximal normal subgroup of | #72326260, #72326261, #72326270, … |
| maximal quotient of | #72328482, #72330334, #72330338 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215326);