Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #197339 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #34633 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,7,6,3,4,8)(9,10,14,18,15,19,13,29)(11,20,24,31,23,35,22,16)(12,25,28,33,26,17,27,32)(21,34,38,30,36,40,37,39);(1,3,6,2)(4,8,5,7)(9,11)(10,16,19,31)(12,21)(13,22)(14,24)(15,23)(17,30,25,39)(18,20,29,35)(26,36)(27,37)(28,38)(32,40,33,34);(9,12)(10,17)(11,21)(13,28)(14,27)(15,26)(16,30)(18,33)(19,25)(20,34)(22,38)(23,36)(24,37)(29,32)(31,39)(35,40);(1,4,6,5)(2,7,3,8)(12,26)(17,25)(21,36)(27,28)(30,39)(32,33)(34,40)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=i,e^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285449, #72286278, #72286708, … |
| maximal normal subgroup of | #72285449, #72286278, #72286708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 104746);