Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2044 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#3199 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1115 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)(5,6,10,16,11,13,9,15)(7,17,21,33,20,34,18,32)(8,22,24,31,23,14,25,30)(12,26,29,19,28,35,27,36);(5,7)(6,12)(8,19)(9,18)(10,21)(11,20)(13,28)(14,17)(15,29)(16,27)(22,34)(23,36)(24,35)(25,26)(30,32)(31,33);(2,4)(6,13)(7,18)(9,10)(12,27)(17,32)(19,35)(20,21)(24,25)(26,36)(28,29)(30,31)(33,34)Polycyclic presentation:
2^9|a^2=g,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,e^b=ehi,f^b=fi,g^b=gh,d^c=dh,f^c=fi,g^c=gi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312042, #72312079, #72312082, … |
| maximal normal subgroup of | #72312042, #72312079, #72312082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55051);