Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,17)(18,20);(5,7,9,8)(6,10,12,11)(14,17)(18,19)(21,22)(23,27)(24,26)(25,28);(5,7,9,8)(6,11,12,10)(13,15,16,20)(14,18,17,19)(22,26)(23,24)(27,28);(5,8,9,7)(6,11,12,10)(14,17)(18,19)(21,23,25,24)(22,27,28,26)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=efi,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268750);