Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13350 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#3027 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,12)(8,13,19,11)(9,21,22,14)(10,15)(16,27,31,35)(17,23,33,26)(18,28)(20,24)(25,30)(29,34)(32,36);(6,11)(7,16)(8,19)(12,25)(13,15)(14,24)(17,32)(18,31)(20,34)(21,29)(23,27)(26,35)(28,36)(30,33);(2,4)(5,7)(6,12)(8,17)(9,16)(10,18)(11,23)(13,26)(14,27)(15,28)(19,33)(20,30)(21,35)(22,31)(24,25)(29,36)(32,34)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72280569, #72280636, #72280814, … |
| maximal normal subgroup of | #72280569, #72280636, #72280814, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54960);