Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632068 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,28)(19,30,34,24)(20,36,38,26)(21,29,23,27)(25,39,44,37)(31,45,32,41)(33,42,35,43)(40,47,46,48);(1,3)(2,5)(4,7)(6,8)(10,13)(11,15)(17,19)(18,24)(20,31)(21,33)(22,34)(23,35)(25,40)(26,45)(27,42)(28,30)(29,43)(32,38)(36,41)(37,47)(39,48)(44,46);(3,7)(5,8)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,31)(21,37)(22,38)(23,39)(24,40)(26,29)(27,36)(28,44)(30,46)(32,34)(33,47)(35,48)(41,42)(43,45);(3,7)(5,8)(10,13)(14,16)(17,20)(18,26)(19,32)(21,39)(22,38)(23,37)(24,41)(25,29)(27,44)(28,36)(30,45)(31,34)(33,47)(35,48)(40,42)(43,46);(2,6)(5,8)(17,21,22,23)(18,27,28,29)(19,33,34,35)(20,37,38,39)(24,42,30,43)(25,36,44,26)(31,47,32,48)(40,41,46,45)Polycyclic presentation:
2^9|a^2=i,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,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, 6539584);