Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(14,20)(21,28)(22,25)(23,24);(2,3)(13,14)(15,25)(16,21)(17,24)(18,20)(19,23)(22,26)(27,28);(1,2,4,3)(6,9)(10,12)(13,15)(14,21)(16,17)(18,26)(19,27)(20,28)(22,24)(23,25);(1,3,4,2)(13,16,18,27)(14,22,20,25)(15,17,26,19)(21,24,28,23);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,20,24)(15,16,26,27)(21,25,28,22)Polycyclic presentation:
2^9|a^2=fi,c^2=fi,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363576);