Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394113 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,21)(9,15)(10,14)(11,16)(12,19)(13,22)(18,26)(20,27)(23,29)(24,28)(25,30)(31,33)(32,35)(34,36);(5,7)(6,12)(8,18)(9,24)(10,19)(11,20)(13,26)(14,27)(15,17)(16,28)(21,35)(22,33)(23,34)(25,32)(29,31)(30,36);(2,4)(5,6)(7,17)(8,13)(9,14)(10,15)(11,16)(12,24)(18,31)(19,28)(20,27)(21,22)(23,30)(25,29)(26,33)(32,36)(34,35);(5,8,11,23)(6,13,16,30)(7,18,20,34)(9,22,10,25)(12,26,28,36)(14,21,15,29)(17,31,27,35)(19,32,24,33)Polycyclic presentation:
2^9|d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,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, 301310);