Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| 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 | #24455 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1503 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18)(19,24)(20,26)(21,27)(22,37)(23,28)(25,34)(29,44)(30,39)(31,40)(32,47)(33,41)(35,42)(36,43)(38,45)(46,48);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,24)(20,29)(21,31)(22,32)(23,33)(25,38)(26,44)(27,40)(28,41)(30,35)(34,45)(36,46)(37,47)(39,42)(43,48);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,38)(26,37)(27,42)(28,43)(30,33)(31,45)(32,46)(39,40)(41,48)(44,47);(2,6)(3,7)(9,11,13,12)(10,16,14,15)(17,20)(18,26)(19,30)(21,36)(22,34)(23,35)(24,39)(25,37)(27,43)(28,42)(29,33)(31,45)(32,46)(38,40)(41,44)(47,48);(3,7)(5,8)(9,12,13,11)(10,16,14,15)(17,21,23,22)(18,27,28,37)(19,31,33,32)(20,34,35,36)(24,40,41,47)(25,42,43,26)(29,45,30,46)(38,39,48,44)Polycyclic presentation:
2^9|c^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7114155);