Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4)(6,7)(10,13)(14,16)(17,18,22,24,21,26,20,25)(19,23,27,32,28,30,29,31);(5,6,8,7)(9,10,12,13)(11,14,15,16)(17,19,22,27,21,28,20,29)(18,23,24,32,26,30,25,31);(2,4)(6,7)(11,15)(14,16)(18,24)(19,27)(20,22)(23,30)(25,26)(28,29);(5,7,8,6)(9,11,12,15)(10,14,13,16)(19,28)(23,30)(27,29)(31,32)Polycyclic presentation:
2^9|a^2=fi,b^2=eghi,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379191);