Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3026 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)(9,10,14,18,15,19,13,23)(11,20,21,24,22,16,12,17);(5,6)(7,8)(9,10)(11,17)(12,20)(13,18)(14,23)(15,19)(16,21)(22,24);(1,2,3,4)(6,8)(9,11,14,21,15,22,13,12)(10,16,18,17,19,20,23,24);(6,8)(9,12,14,11,15,21,13,22)(10,17,18,20,19,24,23,16);(9,13,15,14)(10,18,19,23)(11,12,22,21)(16,17,20,24)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72308735, #72308770, #72308830, … |
| maximal normal subgroup of | #72308735, #72308770, #72308830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351558);