Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3155 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6,11,16,12,7,10,15)(8,19,23,17,21,29,22,13)(9,14,26,24,27,18,25,30)(20,32,35,31,33,36,34,28);(1,3,4,2)(5,7,12,6)(8,17,21,13)(9,24,27,30)(10,15,11,16)(14,25,18,26)(19,22,29,23)(20,32,33,36)(28,34,31,35);(5,8,10,22,12,21,11,23)(6,13,16,19,7,17,15,29)(9,20,25,34,27,33,26,35)(14,28,24,32,18,31,30,36);(5,9,11,26,12,27,10,25)(6,14,16,24,7,18,15,30)(8,20,23,35,21,33,22,34)(13,28,19,32,17,31,29,36);(8,21)(13,17)(19,29)(20,33)(22,23)(28,31)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=g,d^2=fi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,f^b=fi,h^b=hi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279230, #72285921, #72285959, … |
| maximal normal subgroup of | #72279230, #72285921, #72285959, … |
| maximal quotient of | #72279535, #72309543, #72322204 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290010);