Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34935 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3616 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)(5,6)(7,10)(8,9)(11,12)(15,22)(18,27)(23,28)(24,26);(2,4)(13,14)(15,23)(16,20)(17,19)(18,24)(21,25)(22,28)(26,27);(6,9)(10,12)(13,15,16,22)(14,18,20,27)(17,26,25,24)(19,23,21,28);(5,7,8,11)(6,10,9,12)(14,19)(17,25)(18,23)(20,21)(24,26)(27,28);(14,20)(18,27)(19,21)(23,28)Polycyclic presentation:
2^9|c^2=i,d^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72257755, #72281805, #72310306, … |
| maximal normal subgroup of | #72257755, #72281805, #72310306, … |
| maximal quotient of | #72257959, #72316467 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355802);