Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2423 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)(8,12)(9,22)(10,14)(11,17)(13,25)(16,33)(18,31)(19,36)(20,32)(21,30)(23,26)(24,29)(27,35)(28,34);(5,7)(6,11)(8,19)(9,23)(10,18)(12,28)(13,31)(14,26)(15,22)(16,20)(17,25)(21,35)(24,34)(27,29)(30,33)(32,36);(11,26)(15,31)(27,36)(28,33);(2,4)(5,8,10,21)(6,12,14,30)(7,16,18,34)(9,24,25,20)(11,27,26,36)(13,32,22,29)(15,33,31,28)(17,35,23,19)Polycyclic presentation:
2^9|d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72256257, #72256290, #72256515, … |
| maximal normal subgroup of | #72256257, #72256290, #72256515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324867);