Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417541 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1454 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)(7,16,20,27)(8,13)(9,14)(10,25)(11,15)(12,19,28,18)(17,32,34,29)(21,35,33,26)(22,30)(23,31)(24,36);(5,7,9,18,11,20,10,19)(6,12,14,27,15,28,25,16)(8,21,23,34,22,33,24,17)(13,29,31,26,30,32,36,35);(5,8)(6,13)(7,17,20,34)(9,23)(10,24)(11,22)(12,26,28,35)(14,31)(15,30)(16,29,27,32)(18,21,19,33)(25,36);(2,4)(7,18,20,19)(8,22)(12,27,28,16)(13,30)(17,33,34,21)(23,24)(26,29,35,32)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285161, #72285571, #72285668 |
| maximal normal subgroup of | #72285161, #72285571, #72285668 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339372);