Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
5695 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)(9,10)(11,16)(12,14)(13,15)(18,22)(20,23);(9,11,13,12)(10,14,15,16)(17,18)(19,23)(20,24)(21,22);(5,6)(7,8)(17,19)(18,20)(21,24)(22,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24);(9,12,13,11)(10,14,15,16)(17,20,21,23)(18,24,22,19)Polycyclic presentation:
2^9|b^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250100, #72256739, #72256740, … |
| maximal normal subgroup of | #72250100, #72256739, #72256740, … |
| maximal quotient of | #72250296, #72329925, #72330236, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4325677);