Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185491 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6,5,7,3,8)(9,10)(11,19,22,15)(12,23,25,16)(13,17)(14,18)(20,28)(21,31,26,29)(24,36,38,30)(27,32)(33,35)(34,39)(37,40);(2,6)(3,4)(7,8)(9,11,14,22)(10,15,18,19)(12,20,25,35)(13,26,27,21)(16,28,23,33)(17,31,32,29)(24,37,38,34)(30,40,36,39);(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,30)(18,23)(19,33)(21,34)(22,35)(26,37)(27,38)(29,39)(31,40)(32,36);(2,7)(6,8)(9,13,14,27)(10,17,18,32)(11,21,22,26)(12,24,25,38)(15,29,19,31)(16,30,23,36)(20,34,35,37)(28,39,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=ghi,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306032, #72306056, #72306428, … |
| maximal normal subgroup of | #72306032, #72306056, #72306428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92857);