Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394034 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3916 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)(5,6,10,15)(7,16,20,26)(8,13,23,30)(9,12,24,14)(11,17,27,19)(18,33,35,31)(21,29,22,28)(25,34,36,32);(5,7)(6,11)(8,18)(9,17)(10,20)(12,16)(13,25)(14,26)(15,27)(19,24)(21,34)(22,32)(23,35)(28,31)(29,33)(30,36);(2,4)(6,12)(7,17)(9,24)(13,28)(14,15)(16,26)(18,32)(19,20)(21,22)(29,30)(31,33)(34,35);(5,8)(6,13)(7,18)(9,22)(10,23)(11,25)(12,28)(14,29)(15,30)(16,31)(17,32)(19,34)(20,35)(21,24)(26,33)(27,36)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301232);