Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2011 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(10,13)(14,15)(17,18)(19,22)(20,23)(21,24);(5,6)(7,8)(9,10,12,13)(11,15,16,14);(2,4)(6,8)(18,22)(19,20)(23,24);(9,11,12,16)(10,14,13,15)(18,23)(19,20)(22,24)Polycyclic presentation:
2^9|b^2=e,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72305499, #72305516, #72305530, … |
| maximal normal subgroup of | #72305499, #72305516, #72305530, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325778);