Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422715 |
| 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×#3155 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,17)(9,10)(13,27)(14,26)(15,16)(18,29)(22,30)(23,25)(28,32)(33,36)(34,35);(1,2)(3,4)(5,6,10,15,11,16,9,12)(7,18,21,29,17,27,20,13)(8,22,25,14,24,26,23,30)(19,33,35,32,31,28,34,36);(2,4)(5,7,9,21,11,17,10,20)(6,13,12,18,16,29,15,27)(8,19,25,34,24,31,23,35)(14,28,26,32,30,33,22,36);(5,8)(6,14)(7,19)(9,25)(10,23)(11,24)(12,26)(13,28)(15,22)(16,30)(17,31)(18,32)(20,35)(21,34)(27,36)(29,33)Polycyclic presentation:
2^9|b^2=ei,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330438);