Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394823 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1240 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)(7,16)(8,17)(9,23)(10,14)(11,15)(12,19)(13,25)(18,27)(20,28)(21,30)(22,26)(24,29)(31,33)(32,35)(34,36);(5,7,17,20,11,21,10,19)(6,12,14,30,15,28,8,16)(9,18,25,33,26,34,24,32)(13,27,23,35,29,36,22,31);(1,2,4,3)(5,8,17,6,11,14,10,15)(7,12,20,30,21,28,19,16)(9,22,25,13,26,23,24,29)(18,27,33,35,34,36,32,31);(1,3,4,2)(5,9,11,26)(6,13,15,29)(7,18,21,34)(8,22,14,23)(10,24,17,25)(12,27,28,36)(16,31,30,35)(19,32,20,33)Polycyclic presentation:
2^9|b^2=ei,c^2=fi,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 308483);