Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #24271 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,21,26,36,24)(19,28,31,38,30,23,37,40)(20,32,35,41,34,25,42,44)(29,39,45,48,33,43,46,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,33)(21,30)(22,31)(24,38)(25,43)(26,28)(27,40)(29,34)(32,39)(35,46)(36,37)(41,48)(42,45)(44,47);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(18,24)(22,36)(23,38)(25,41)(26,27)(28,40)(31,37)(32,44)(35,42)(39,47)(43,48)(45,46);(2,6)(7,8)(11,16)(13,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,39)(24,41)(26,32)(27,44)(28,43)(30,33)(31,45)(36,42)(37,46)(38,47)(40,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(19,30)(20,34)(23,28)(25,32)(31,37)(35,42)(38,40)(41,44)Polycyclic presentation:
2^9|a^2=g,c^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6919928);