Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457579 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15,18,11)(8,12)(9,13)(10,14)(16,23,19,29)(17,30,32,24)(20,27)(21,26)(22,28)(25,34)(31,36,33,35);(1,2)(3,4)(5,7)(6,11)(8,19)(9,17)(10,18)(12,23)(13,24)(14,15)(16,21)(20,33)(22,32)(25,35)(26,29)(27,36)(28,30)(31,34);(2,4)(5,8)(6,12)(7,16,18,19)(9,20)(10,21)(11,23,15,29)(13,27)(14,26)(17,31,32,33)(22,34)(24,36,30,35)(25,28);(5,9,10,22)(6,13,14,28)(7,17,18,32)(8,20,21,34)(11,24,15,30)(12,25,26,27)(16,31,19,33)(23,35,29,36)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72240509, #72283596, #72283644, … |
| maximal normal subgroup of | #72240509, #72283596, #72283644, … |
| maximal quotient of | #72235753, #72266477, #72266478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364776);