Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 | |
1506 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)(5,6)(7,8)(9,10,13,12)(11,15,16,14)(18,22)(20,24)(23,27)(25,31)(26,32)(28,30);(9,11,13,16)(10,14,12,15)(17,18,20,22,21,25,24,31)(19,26,28,32,29,27,30,23);(2,4)(10,12)(14,15)(17,19,21,29)(18,23,25,32)(20,30,24,28)(22,27,31,26);(6,8)(9,12,13,10)(11,15,16,14)(17,18,24,31,21,25,20,22)(19,27,30,32,29,26,28,23)Polycyclic presentation:
2^9|a^2=egi,b^2=e,c^2=i,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72264195, #72264198, #72264257, … |
| maximal normal subgroup of | #72264195, #72264198, #72264257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367577);