Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #34980 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4809 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,10,14,18)(8,15,16,11)(9,12)(13,17)(19,20);(5,7)(6,10,12,18)(8,13)(9,14)(11,17,15,20)(16,19);(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,19)(18,20);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=eh,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,f^b=fi,h^b=hi,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72213862, #72218088, #72219038, … |
| maximal normal subgroup of | #72213862, #72218088, #72219038, … |
| maximal quotient of | #72210458, #72213628, #72234596, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299895);