Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45316 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,26)(19,27,31,45)(20,33,35,25)(21,38,32,24)(22,40,29,28)(30,43,34,46)(36,42,39,41)(37,44,47,48);(2,5)(3,8)(9,11)(10,13)(12,16)(14,15)(17,19,29,21,23,31,22,32)(18,24,40,45,26,38,28,27)(20,34,47,36,35,30,37,39)(25,42,48,46,33,41,44,43);(1,3,4,8)(2,6,5,7)(10,14)(11,16)(17,20)(18,25)(19,30)(21,39)(22,37)(23,35)(24,41)(26,33)(27,46)(28,44)(29,47)(31,34)(32,36)(38,42)(40,48)(43,45);(2,5)(6,7)(10,14)(11,16)(18,26)(19,31)(20,35)(21,32)(28,40)(37,47)(41,42)(43,46);(2,5)(6,7)(10,14)(13,15)(17,21,29,31,23,32,22,19)(18,27,40,24,26,45,28,38)(20,36,47,30,35,39,37,34)(25,43,48,42,33,46,44,41)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=gh,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7508149);