Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2620 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)(7,17)(8,23)(9,16)(10,14)(11,15)(12,20)(13,24)(18,27)(19,35)(21,28)(22,29)(25,31)(26,30)(32,34)(33,36);(1,2,3,4)(5,7,9,20,11,21,10,22)(6,12,14,17,15,29,16,28)(8,19,24,36,25,18,26,34)(13,27,23,33,30,35,31,32);(5,8)(6,13)(7,18)(9,26)(10,24)(11,25)(12,27)(14,31)(15,30)(16,23)(17,32)(19,21)(20,36)(22,34)(28,33)(29,35);(5,8)(6,13)(7,19)(9,24)(10,26)(11,25)(12,27)(14,23)(15,30)(16,31)(17,33)(18,21)(20,36)(22,34)(28,32)(29,35)Polycyclic presentation:
2^9|b^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72341354, #72341355, #72341366, … |
| maximal normal subgroup of | #72341354, #72341355, #72341366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301578);