Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
991 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,8)(7,11,12,10)(13,14,18,22)(15,23,25,20)(16,26,24,28)(17,21,19,27);(5,7)(6,10)(8,11)(9,12)(13,15,19,24,18,25,17,16)(14,20,27,26,22,23,21,28);(5,6,9,8)(7,10,12,11)(13,16,19,25,18,24,17,15)(14,20,21,26,22,23,27,28);(2,4)(5,8,9,6)(7,10,12,11)(13,17,18,19)(14,21,22,27)(15,24,25,16)(20,26,23,28)Polycyclic presentation:
2^9|a^2=e,b^2=fg,c^2=efi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 346457);