Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| 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 | |
2782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,15,12)(11,16,13,17)(14,18,24,22)(19,20,23,21);(1,2,3,4)(9,11,15,13)(10,16,12,17)(14,20,24,21)(18,23,22,19);(6,8)(9,12,15,10)(11,19,13,23)(14,18,24,22)(16,20,17,21);(9,13,15,11)(10,17,12,16)(14,21,24,20)(18,19,22,23);(11,20)(13,21)(14,24)(16,23)(17,19)(18,22)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72285147 |
| maximal normal subgroup of | #72285147 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481046);