Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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 | C2×#3460 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3355 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)(8,11)(9,12)(13,14)(15,23)(16,21)(17,19)(18,22)(20,24)(25,27)(26,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(14,19)(15,24)(16,17)(20,28)(21,22)(23,27);(2,4)(5,7,9,8)(6,11,12,10)(13,15,18,24)(14,20,22,23)(16,25,17,26)(19,27,21,28);(2,4)(5,8,9,7)(6,11,12,10)(13,16,18,17)(14,21,22,19)(15,25,24,26)(20,28,23,27);(15,24)(20,23)(25,26)(27,28)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72305349, #72305359, #72305366 |
| maximal normal subgroup of | #72305349, #72305359, #72305366 |
| maximal quotient of | #72305622, #72305627 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360107);