Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,24,25,20)(16,26,27,21)(17,22,19,28);(1,3,4,2)(6,9)(7,11)(13,15,18,25)(14,20,23,24)(16,17,27,19)(21,22,26,28);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,19)(17,25)(18,27)(21,22)(23,24)(26,28);(6,9)(10,12)(13,16)(14,21)(15,17)(18,27)(19,25)(20,22)(23,26)(24,28);(13,17,18,19)(14,22,23,28)(15,16,25,27)(20,21,24,26)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^b=ei,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282860);