Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| 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 | |
1584 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)(8,11)(9,23)(10,15)(12,16)(13,19)(14,26)(18,32)(20,29)(21,28)(22,24)(25,30)(27,34)(31,35)(33,36);(5,7,10,19,12,21,11,20)(6,13,8,17,16,29,15,28)(9,18,24,33,26,35,25,34)(14,27,22,31,23,36,30,32);(1,2,3,4)(5,8,10,16,12,15,11,6)(7,13,19,17,21,29,20,28)(9,22,24,23,26,30,25,14)(18,27,33,31,35,36,34,32);(5,9,10,24,12,26,11,25)(6,14,8,22,16,23,15,30)(7,18,19,33,21,35,20,34)(13,27,17,31,29,36,28,32)Polycyclic presentation:
2^9|b^2=e,c^2=fi,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 305647);