Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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×#3543 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3422 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)(13,14)(15,23)(16,21)(17,19)(18,22)(20,25)(24,28)(26,27);(2,4)(5,6)(7,10)(8,9)(11,12)(14,19)(16,17)(20,27)(21,22)(23,28)(24,25);(6,9)(10,12)(13,15)(14,20)(16,24)(17,25)(18,26)(19,27)(21,23)(22,28);(5,7)(6,10)(8,11)(9,12)(13,16,18,17)(14,21,22,19)(15,24,26,25)(20,23,28,27);(7,11)(10,12)Polycyclic presentation:
2^9|a^2=fg,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72308081, #72308103, #72308427 |
| maximal normal subgroup of | #72308081, #72308103, #72308427 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449903);