Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,8,9)(7,11,12,10)(13,14)(15,24)(16,27)(17,20)(18,22)(19,23)(21,30)(25,35)(26,40)(28,37)(29,42)(31,41)(32,36)(33,39)(34,38)(43,44);(6,9)(7,12)(13,15)(14,20)(16,28)(17,18)(19,32)(21,37)(22,24)(23,40)(25,31)(26,33)(27,44)(29,34)(30,43)(35,38)(36,39)(41,42);(2,4)(13,16,18,30)(14,21,22,27)(15,25,17,29)(19,34,33,31)(20,35,24,42)(23,41,39,38)(26,43,32,28)(36,44,40,37);(2,4)(5,7,8,12)(6,10,9,11)(13,17,18,15)(14,20,22,24)(16,29,30,25)(19,32,33,26)(21,35,27,42)(23,36,39,40)(28,31,43,34)(37,41,44,38);(14,22)(15,17)(21,27)(23,39)(25,29)(26,32)(28,43)(38,41)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409446);