Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,11,9,7,10,12)(13,14,17,21,18,22,16,20)(15,23,24,19,26,27,25,28);(1,2,3,4)(5,7,8,12,9,6,10,11)(13,15)(14,19)(16,24)(17,25)(18,26)(20,27)(21,23)(22,28);(13,15)(14,19)(16,24)(17,25)(18,26)(20,27)(21,23)(22,28);(6,7)(11,12)(13,16,18,17)(14,20,22,21)(15,24,26,25)(19,27,28,23)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72310300, #72310301, #72310318 |
| maximal normal subgroup of | #72310300, #72310301, #72310318 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100846);