Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60750 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
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×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3682 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)(3,4)(6,9)(7,11)(13,14,18,21,17,19,16,22)(15,23,25,20,24,28,26,27);(5,6)(7,10)(8,9)(11,12)(14,19)(15,24)(21,22)(25,26);(2,4)(6,9)(10,12)(13,15)(14,20)(16,25)(17,24)(18,26)(19,27)(21,23)(22,28);(5,7,8,11)(6,10,9,12)(14,19)(20,27)(21,22)(23,28);(13,16,17,18)(14,21,19,22)(15,25,24,26)(20,23,27,28)Polycyclic presentation:
2^9|a^2=h,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cf,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72240281, #72240283, #72240286, … |
| maximal normal subgroup of | #72240281, #72240283, #72240286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281648);