Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15947 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(5,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22);(10,14)(15,16)(18,22)(19,24);(1,2)(3,4)(5,6)(7,8)(9,11,13,12)(10,15,14,16)(17,19,21,24)(18,23,22,20);(5,6)(7,8)(9,12,13,11)(10,15,14,16)(17,20)(18,19)(21,23)(22,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219266, #72258999, #72260512, … |
| maximal normal subgroup of | #72219259, #72219266, #72258999, … |
| maximal quotient of | #72298180, #72323621, #72323631, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1329218);