Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1482 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,26)(19,27,30,31)(20,21,25,22)(24,28,32,29);(5,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,31)(21,28)(22,29)(23,30)(25,27)(26,32);(2,4)(6,8)(17,20,21,26,23,25,22,18)(19,24,29,27,30,32,28,31);(10,14)(13,16)(17,21,23,22)(18,20,26,25)(19,28,30,29)(24,31,32,27)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264775, #72264787, #72264835, … |
| maximal normal subgroup of | #72264775, #72264787, #72264835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312598);