Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,17,30)(8,13)(9,14)(10,12)(11,19,26,23)(16,24)(18,28,32,36)(20,33,34,25)(21,29)(22,27)(31,35);(5,7,16,19,10,17,9,23)(6,11,24,15,12,26,14,30)(8,20,31,32,22,34,21,18)(13,28,35,33,27,36,29,25);(6,12)(7,17)(9,16)(13,27)(15,30)(18,32)(21,31)(28,36);(2,4)(5,8,10,22)(6,13,12,27)(7,18)(9,21,16,31)(11,25)(14,29,24,35)(15,28)(17,32)(19,20)(23,34)(26,33)(30,36)Polycyclic presentation:
2^9|a^2=fi,b^2=ef,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72283599, #72283686, #72283699, … |
| maximal normal subgroup of | #72283599, #72283686, #72283699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337196);