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 | |
3470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,18,22,29)(8,23)(9,15)(10,16)(11,17)(12,21,13,20)(14,26)(19,34,36,31)(24,28,35,30)(25,33)(27,32);(1,2)(3,4)(5,7)(6,12)(8,24)(9,21)(10,20)(11,22)(13,17)(14,31)(15,18)(16,29)(19,25)(23,34)(26,35)(27,36)(28,32)(30,33);(2,4)(5,7)(6,13)(8,19)(9,20)(10,21)(11,22)(12,17)(14,30)(15,18)(16,29)(23,28)(24,25)(26,36)(27,35)(31,33)(32,34);(5,8)(6,14)(7,19)(9,25)(10,27)(11,26)(12,28)(13,30)(15,32)(16,33)(17,23)(18,34)(20,24)(21,35)(22,36)(29,31);(5,9,11,10)(6,15,17,16)(7,20,22,21)(8,25,26,27)(12,29,13,18)(14,32,23,33)(19,24,36,35)(28,31,30,34)Polycyclic presentation:
2^9|a^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,f^b=fi,g^c=gi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410572);