Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187766 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,8,5,6,3,7)(9,10)(11,19,22,31)(12,23,25,16)(13,17)(14,18)(15,21,29,26)(20,34)(24,38,40,30)(27,32)(28,35)(33,36)(37,39);(2,6)(3,4)(9,11,14,22)(10,15,18,29)(12,20,25,36)(13,26,27,21)(16,28,23,37)(17,31,32,19)(24,39,40,35)(30,34,38,33);(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,30)(18,23)(19,33)(21,35)(22,36)(26,39)(27,40)(29,37)(31,34)(32,38);(1,3,5,4)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=ghi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306308, #72306322, #72306368, … |
| maximal normal subgroup of | #72306308, #72306322, #72306368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95025);