Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417542 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2162 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)(7,11)(8,18,21,12)(9,13)(10,14)(15,28,19,23)(16,24)(17,25)(20,31,33,26)(22,27)(29,36,32,34)(30,35);(5,7)(6,11)(8,19,21,15)(9,16)(10,17)(12,23,18,28)(13,24)(14,25)(20,29,33,32)(22,30)(26,36,31,34)(27,35);(5,8)(6,12)(7,15)(9,20)(10,21)(11,23)(13,26)(14,18)(16,29)(17,19)(22,33)(24,34)(25,28)(27,31)(30,32)(35,36);(2,4)(5,9,10,22)(6,13,14,27)(7,16,17,30)(8,20,21,33)(11,24,25,35)(12,26,18,31)(15,29,19,32)(23,34,28,36)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72210036, #72210037, C2×#417970, … |
| maximal normal subgroup of | #72210036, #72210037, C2×#417970, … |
| maximal quotient of | #72240909, #72240910, #72240914, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325166);