Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 | C2×#120 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,9,11,12)(13,14)(15,23)(16,19)(17,27)(18,20)(21,25)(22,24)(26,28);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,24,26,18,25,17,16)(14,19,27,21,20,28,22,23);(6,10)(7,11)(14,20)(15,25)(16,26)(22,27);(2,4)(6,10)(9,12)(14,20)(15,25)(16,26)(22,27);(13,16,17,25,18,26,24,15)(14,21,22,19,20,23,27,28)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7434758);