Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(13,14)(15,20)(16,23)(17,19)(18,22)(21,25)(24,27)(26,28);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,19,22)(14,18,17,20)(16,23,26,28)(21,25,24,27);(2,4)(13,16)(14,21)(15,23)(17,24)(18,25)(19,26)(20,27)(22,28);(5,7,8,11)(6,10,9,12)(13,17,19,14)(15,18,22,20)(16,24,26,21)(23,25,28,27);(2,4)(6,9)(10,12)(13,18,19,20)(14,22,17,15)(16,25,26,27)(21,28,24,23)Polycyclic presentation:
2^9|b^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
| maximal subgroup of | #72283280, #72283706, #72284213, … |
| maximal normal subgroup of | #72283280, #72283706, #72284213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579113);