Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394353 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1622 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)(7,13)(8,21)(9,23)(10,16)(11,15)(12,17)(14,24)(18,32)(19,28)(20,29)(22,35)(25,30)(26,31)(27,33)(34,36);(1,2,4,3)(5,7,12,20)(6,13,17,29)(8,10,19,11)(9,18,25,34)(14,27,31,35)(15,21,16,28)(22,24,33,26)(23,32,30,36);(5,8,11,20,12,19,10,7)(6,13,16,28,17,29,15,21)(9,22,26,34,25,33,24,18)(14,27,30,36,31,35,23,32);(1,3,4,2)(5,9)(6,14)(7,18)(8,22)(10,24)(11,26)(12,25)(13,27)(15,23)(16,30)(17,31)(19,33)(20,34)(21,32)(28,36)(29,35)Polycyclic presentation:
2^9|b^2=ei,c^2=fi,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305565);