Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#1999 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6)(7,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,24)(20,22);(1,2)(3,4)(9,11)(10,13)(12,16)(14,15)(17,18)(19,20)(21,24)(22,23);(2,4)(10,14)(13,15)(18,24)(19,23);(6,8)(11,16)(13,15)(17,19,22,24,21,23,20,18)Polycyclic presentation:
2^9|d^2=efi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72264436, #72264473, #72264584, … |
| maximal normal subgroup of | #72264436, #72264473, #72264584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335053);