Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 |
| prime radical | #38910 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2030 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)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(13,14,19,23)(15,24,25,22)(16,21,18,27)(17,20,26,28);(2,4)(13,15,16,26,19,25,18,17)(14,20,27,22,23,28,21,24);(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,21,23,27)(15,17,25,26)(20,22,28,24);(6,9)(10,12)(13,17,18,25,19,26,16,15)(14,22,21,20,23,24,27,28)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256862, #72256890, #72257047 |
| maximal normal subgroup of | #72256862, #72256890, #72257047 |
| maximal quotient of | #72264281, #72264880 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715409);