Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #42443 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1925 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)(5,6)(7,10)(8,12)(9,11)(13,14,17,19)(15,16,20,24)(18,21,26,28)(22,25,27,23);(5,7,10,6)(8,12,9,11)(14,19)(15,22)(16,23)(18,26)(20,27)(24,25);(2,4)(13,15,17,20)(14,16,19,24)(18,27,26,22)(21,23,28,25);(5,8,10,9)(6,11,7,12)(14,19)(16,24)(21,28)(23,25);(5,9,10,8)(6,11,7,12)(13,16)(14,20)(15,19)(17,24)(18,23)(21,22)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584071);