Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,27,23,28,22,24)(19,25,20,26,30,32,29,31);(1,2,3,4)(18,24)(19,29)(20,30)(21,22)(25,32)(27,28);(2,4)(9,11,12,16)(10,13,14,15)(17,19,22,29,23,30,21,20)(18,25,24,31,28,32,27,26);(6,8)(17,20,22,19,23,29,21,30)(18,26,24,25,28,31,27,32)Polycyclic presentation:
2^9|a^2=ef,b^2=eg,c^2=efi,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 346155);