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×#3062 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(14,18)(16,23)(19,24)(20,25)(22,27)(26,28);(1,3,4,2)(5,7)(6,9)(8,12)(10,11);(13,14,16,18,17,20,23,25)(15,19,22,24,21,26,27,28);(7,12)(9,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(6,10)(9,11)(14,20)(15,21)(18,25)(22,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^c=ei,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72308623, #72308831 |
| maximal normal subgroup of | #72308623, #72308831 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6296418);