Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #37559 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,34)(28,37)(29,36)(30,38)(31,40)(32,42)(33,39)(35,43)(41,46)(44,48)(45,47);(1,2)(3,6)(4,8)(5,7)(10,13)(11,15)(17,19,22,28)(18,23,26,37)(20,31,32,29)(21,30,34,43)(24,38,27,35)(25,40,42,36)(33,45,46,44)(39,47,41,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(18,24)(19,28)(21,34)(23,35)(25,39)(26,27)(29,31)(33,46)(36,47)(37,38)(40,48)(41,42);(2,7)(6,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,33)(22,32)(23,36)(24,39)(26,42)(27,41)(28,31)(30,44)(34,46)(35,47)(37,40)(38,48)(43,45);(1,4,5,3)(2,8,7,6)(18,26)(19,28)(20,32)(24,27)(30,43)(33,46)(36,40)(47,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,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, 6561249);