Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427374 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15,18,11)(8,19,22,12)(9,13)(10,14)(16,29)(17,25,31,27)(20,24)(21,32,34,26)(23,28)(30,35)(33,36);(2,4)(5,7)(6,11)(8,20)(9,17)(10,18)(12,24)(13,27)(14,15)(16,22)(19,29)(21,33)(23,31)(25,28)(26,36)(30,34)(32,35);(1,2,3,4)(5,8)(6,12)(7,16,18,20)(9,21)(10,22)(11,24,15,29)(13,26)(14,19)(17,30,31,33)(23,34)(25,35,27,36)(28,32);(5,9,10,23)(6,13,14,28)(7,17,18,31)(8,21,22,34)(11,25,15,27)(12,26,19,32)(16,30,20,33)(24,35,29,36)Polycyclic presentation:
2^9|a^2=fg,c^2=ef,d^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72283578, #72283596, #72283846, … |
| maximal normal subgroup of | #72283578, #72283596, #72283846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341913);