Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| 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×#3355 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1853 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)(7,11)(8,10)(9,12)(13,14)(15,21)(16,19)(17,26)(18,25)(20,23)(22,27)(24,28);(5,7)(6,9)(8,12)(10,11)(13,14,16,19)(15,22,24,18)(17,20,23,26)(21,25,28,27);(13,15,23,27,16,24,17,25)(14,18,26,28,19,22,20,21);(7,12)(9,11)(13,14)(15,22)(16,19)(17,20)(18,24)(21,27)(23,26)(25,28);(1,3,4,2)(6,10)(9,11)(14,19)(15,24)(20,26)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,d^c=di,e^c=ei,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6310520);