Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1827 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,27,30)(23,25,29,26)(24,32,33,31)(28,34,36,35);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,30,27,29,26,22)(24,32,34,28,33,31,35,36);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24,25,34,27,33,26,35)(22,28,23,31,30,36,29,32);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(24,33)(28,36)(31,32)(34,35);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,29,30,23)(24,34,33,35)(28,32,36,31)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307833, #72307977, #72308579, … |
| maximal normal subgroup of | #72307833, #72307977, #72308579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6473533);