Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23900 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×B(4,𝔽2) | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(5,6,8,7)(13,16);(1,2)(3,4)(5,7,8,6)(9,10)(11,13)(12,14)(15,16);(2,4)(9,11)(10,13)(12,15)(14,16);(9,10)(11,13)(12,14)(15,16)Polycyclic presentation:
2^9|a^2=h,b^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216660, #72220578, #72232809, … |
| maximal normal subgroup of | #72216660, #72232809, #72232941, … |
| maximal quotient of | #72216620, #72219233, #72233066, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271709);