Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395246 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3154 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,9)(8,21)(10,28)(11,16)(12,17)(13,19)(14,25)(15,26)(18,33)(20,27)(22,30)(23,35)(24,31)(29,32)(34,36);(1,2,4,3)(5,7,10,19,12,20,11,15)(6,13,16,9,17,26,28,27)(8,22,24,18,25,34,29,35)(14,23,31,36,21,33,32,30);(1,3,4,2)(5,8,10,24,12,25,11,29)(6,14,16,31,17,21,28,32)(7,18,19,34,20,35,15,22)(9,23,26,36,27,33,13,30);(5,9,10,26,12,27,11,13)(6,15,16,7,17,19,28,20)(8,23,24,36,25,33,29,30)(14,22,31,18,21,34,32,35)Polycyclic presentation:
2^9|b^2=e,c^2=e,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319369);