Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6,9,13,11,15,10,16)(7,17,18,27,21,28,20,12)(8,14,23,29,25,31,24,32)(19,30,22,35,34,36,33,26);(2,4)(5,7)(6,12)(8,22)(9,20)(10,18)(11,21)(13,28)(14,30)(15,27)(16,17)(19,23)(24,34)(25,33)(26,29)(31,36)(32,35);(6,13)(7,18)(9,10)(14,29)(15,16)(17,28)(19,22)(20,21)(23,24)(30,36)(31,32)(33,34);(5,8)(6,14)(7,19)(9,23)(10,24)(11,25)(12,26)(13,29)(15,31)(16,32)(17,30)(18,22)(20,33)(21,34)(27,35)(28,36);(6,15)(12,27)(13,16)(14,31)(17,28)(26,35)(29,32)(30,36)Polycyclic presentation:
2^9|a^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72256497, #72256516, #72256520, … |
| maximal normal subgroup of | #72256497, #72256516, #72256520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345099);