Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(7,11)(10,12)(13,14)(15,19)(16,18)(17,21)(20,24)(22,26)(23,25)(27,28);(2,3)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(1,3,4,2)(6,9)(10,12)(13,15)(14,19)(16,23)(17,22)(18,25)(20,27)(21,26)(24,28);(5,7,8,11)(6,10,9,12)(15,22)(19,26)(23,28)(25,27)Polycyclic presentation:
2^9|a^2=f,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
| maximal subgroup of | #72282583, #72283797, #72284173, … |
| maximal normal subgroup of | #72282583, #72283797, #72284173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053099);