Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5086 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10)(11,19)(12,18)(13,15)(14,17)(16,20)(21,24)(22,23);(1,2,3,4)(10,15)(16,23)(17,18)(19,24);(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,19)(18,24);(6,8);(9,12,13,14)(10,17,15,18)(11,20,21,22)(16,19,23,24)Polycyclic presentation:
2^9|b^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72281444, #72281515, #72281525, … |
| maximal normal subgroup of | #72281444, #72281515, #72281525, … |
| maximal quotient of | #72306074, #72306105, #72317827 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536126);