Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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 | |
1869 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)(22,26)(23,29)(24,31)(27,34)(28,33)(30,35);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,22,24,26,25,28,31,33)(23,27,29,32,30,34,35,36);(2,4)(7,13)(10,16)(14,19)(17,20)(21,23,24,29,25,30,31,35)(22,27,26,32,28,34,33,36);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(22,28)(26,33)(27,34)(32,36);(8,15)(11,18)(14,19)(17,20)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307490, #72307628, #72307728, … |
| maximal normal subgroup of | #72307490, #72307628, #72307728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6473468);