Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
3182 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)(5,6,9,16,11,15,10,17)(7,18,20,29,22,30,21,12)(8,13,24,32,25,31,26,14)(19,28,23,33,34,36,35,27);(5,7,11,22)(6,12,15,29)(8,23,25,35)(9,21,10,20)(13,28,31,36)(14,33,32,27)(16,30,17,18)(19,26,34,24);(2,4)(5,8)(6,13)(7,19)(9,24)(10,26)(11,25)(12,27)(14,17)(15,31)(16,32)(18,28)(20,23)(21,35)(22,34)(29,33)(30,36);(2,4)(5,8)(6,14)(7,19)(9,26)(10,24)(11,25)(12,28)(13,17)(15,32)(16,31)(18,27)(20,35)(21,23)(22,34)(29,36)(30,33);(6,15)(12,29)(13,31)(14,32)(16,17)(18,30)(27,33)(28,36)Polycyclic presentation:
2^9|a^2=f,b^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,f^b=fi,h^b=hi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250742, #72250779, #72250789, … |
| maximal normal subgroup of | #72250742, #72250779, #72250789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408105);