Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,8)(9,10,13,12)(11,15,16,14)(17,18,21,20)(19,23,24,22);(2,4)(6,8)(18,20)(22,23);(1,2,3,4)(5,6,7,8)(19,24)(22,23);(9,11,13,16)(10,14,12,15)(17,19,21,24)(18,22,20,23);(9,12,13,10)(11,15,16,14)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72213205, #72219260, #72219262, … |
| maximal normal subgroup of | #72210448, #72213205, #72219260, … |
| maximal quotient of | #72250296, #72251025, #72323524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 665412);