Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2867 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)(5,6)(7,8)(9,10,14,18)(11,15,21,24)(12,16,19,22)(13,17,20,23);(1,2,3,4)(9,11,14,21)(10,15,18,24)(12,13,19,20)(16,17,22,23);(6,8)(11,19)(12,21)(13,20)(15,22)(16,24)(17,23);(9,12,14,19)(10,16,18,22)(11,20,21,13)(15,23,24,17);(11,12)(13,20)(15,16)(17,23)(19,21)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285175, #72308822, #72308960, … |
| maximal normal subgroup of | #72285175, #72308822, #72308960, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481072);