Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2850 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)(10,15)(11,19)(17,24)(18,23);(1,2,3,4)(9,10,13,15)(11,20,19,21)(12,17,16,24)(14,23,22,18);(6,8)(9,11,13,19)(10,16,15,12)(14,24,22,17)(18,21,23,20);(9,12)(10,17)(11,18)(13,16)(14,20)(15,24)(19,23)(21,22);(10,18)(12,20)(14,22)(15,23)(16,21)(17,24)Polycyclic presentation:
2^9|b^2=f,c^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250364, #72250399, #72256943, … |
| maximal normal subgroup of | #72250364, #72250399, #72256943, … |
| maximal quotient of | #72264553, #72265178, #72265453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481359);