Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | #31063 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(13,14,18,21)(15,22,20,17)(16,19,25,27)(23,28,24,26);(2,3)(5,6)(7,10)(8,9)(11,12);(1,2,4,3)(13,15,17,14,18,20,22,21)(16,24,26,27,25,23,28,19);(5,7,8,11)(6,10,9,12)(13,16,18,25)(14,19,21,27)(15,23,20,24)(17,26,22,28);(1,3,4,2)(6,9)(10,12)(16,25)(19,27)(23,24)(26,28)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722384);