Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| 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 | #38049 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14,20,23)(15,16,24,26)(17,21,27,25)(18,22,19,28);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,16)(17,19)(18,27)(20,24)(21,28)(22,25)(23,26);(6,9)(7,11)(13,16,19,25,20,26,18,21)(14,17,22,24,23,27,28,15);(5,7)(6,10)(8,11)(9,12)(13,17)(14,21)(15,18)(16,22)(19,24)(20,27)(23,25)(26,28);(6,9)(7,11)(13,18,20,19)(14,22,23,28)(15,17,24,27)(16,21,26,25)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^d=ehi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513726);