Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1701 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)(9,10)(11,14)(12,13)(15,16)(17,18,21,26)(19,28,30,23)(20,32,33,25)(22,24,35,27)(29,37,42,41)(31,38,44,36)(34,40,45,39)(43,46,48,47);(1,3)(2,5)(4,7)(6,8)(10,13)(11,15)(17,19,21,30)(18,23,26,28)(20,29,33,42)(22,31,35,44)(24,36,27,38)(25,37,32,41)(34,43,45,48)(39,46,40,47);(3,7)(5,8)(9,11,12,15)(10,14,13,16)(18,24)(22,35)(23,36)(25,39)(26,27)(28,38)(31,44)(32,40)(34,45)(37,46)(41,47)(43,48);(3,7)(5,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,37)(24,39)(26,32)(27,40)(28,41)(30,42)(31,43)(35,45)(36,46)(38,47)(44,48);(2,6)(5,8)(18,26)(19,30)(20,33)(24,27)(31,44)(34,45)(37,41)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6539623);