Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(9,10,12,17)(11,16,19,14)(13,15,21,18)(20,23,24,22);(1,2,4,3)(5,6)(7,8)(9,11)(10,14)(12,19)(13,20)(15,22)(16,17)(18,23)(21,24);(1,3,4,2)(6,8)(10,15)(13,21)(14,22)(16,23)(17,18)(20,24);(6,8)(9,11)(10,16)(12,19)(13,20)(14,17)(15,23)(18,22)(21,24);(10,17)(11,19)(15,18)(20,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72342359, #72342370, #72342372 |
| maximal normal subgroup of | #72342359, #72342370, #72342372 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346035);