Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2788 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)(9,10,12,18,13,17,21,14)(11,16,20,22,19,15,24,23);(5,6)(7,8)(10,14)(12,21)(15,22)(16,23)(17,18)(20,24);(1,2,3,4)(6,8)(9,11)(10,15)(12,20)(13,19)(14,22)(16,17)(18,23)(21,24);(6,8)(9,11)(10,16)(12,20)(13,19)(14,23)(15,17)(18,22)(21,24);(10,17)(11,19)(14,18)(20,24)Polycyclic presentation:
2^9|a^2=f,c^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72316435, #72316463 |
| maximal normal subgroup of | #72316435, #72316463 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350213);