Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417732 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| 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 | |
1285 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,13)(9,12)(10,15)(11,18)(14,24)(17,32)(19,27)(20,25)(21,35)(22,28)(23,31)(26,33)(29,34)(30,36);(5,7)(6,11)(8,21)(9,19)(10,20)(12,16)(13,29)(14,25)(15,27)(17,22)(18,24)(23,33)(26,30)(28,35)(31,32)(34,36);(6,12)(9,24)(11,25)(13,28)(14,15)(16,27)(18,19)(21,33)(22,36)(26,32)(29,35)(30,31);(2,4)(5,8,10,23)(6,13,15,31)(7,17)(9,22,24,36)(11,26)(12,28,14,30)(16,29)(18,21)(19,33)(20,34)(25,32)(27,35)Polycyclic presentation:
2^9|d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,f^c=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335072);