Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(14,15)(18,19)(21,22)(23,27)(24,26)(25,28);(1,2,3,4)(6,9)(10,12)(13,14,17,15)(16,19,20,18)(21,22,24,26)(23,25,28,27);(1,2)(3,4)(21,23)(22,25)(24,28)(26,27);(5,7,8,11)(6,10,9,12)(13,15,17,14)(16,19,20,18)(21,23)(22,25)(24,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,18)(15,19)(17,20)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|b^2=hi,d^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72267287, #72326256, #72326267, … |
| maximal normal subgroup of | #72267287, #72326256, #72326267, … |
| maximal quotient of | #72267316, #72328549, #72328701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 564274);