Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5894 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)(6,8)(11,12)(15,16)(17,18)(19,23)(20,21)(22,24);(1,2)(3,4)(5,6)(7,8)(9,10,14,13)(11,16,12,15)(18,20)(22,23);(9,11,14,12)(10,15,13,16)(17,19)(18,22)(20,23)(21,24);(5,6)(7,8)(9,12,14,11)(10,16,13,15)(17,20,21,18)(19,23,24,22);(1,2)(3,4)(5,6)(7,8)(9,13,14,10)(11,16,12,15)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72323324, #72323326, #72323334, … |
| maximal normal subgroup of | #72323324, #72323326, #72323334, … |
| maximal quotient of | #72323683, #72330730, #72330785, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 757701);