Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234345 |
| 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 | #31066 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1429 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)(3,4)(17,18,23,27)(19,24,22,36)(20,30,32,38)(21,33,35,26)(25,37,41,28)(29,42,34,39)(31,45,47,44)(40,43,48,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,23,22)(18,24,27,36)(20,28,32,37)(21,34,35,29)(25,38,41,30)(26,42,33,39)(31,46,47,43)(40,44,48,45);(1,3,5,4)(2,7,6,8)(10,13)(11,15)(17,20,23,32)(18,25,27,41)(19,28,22,37)(21,31,35,47)(24,38,36,30)(26,40,33,48)(29,43,34,46)(39,45,42,44);(2,6)(7,8)(9,11,12,15)(10,14,13,16)(17,21,23,35)(18,26,27,33)(19,29,22,34)(20,31,32,47)(24,39,36,42)(25,40,41,48)(28,43,37,46)(30,44,38,45);(1,4,5,3)(2,8,6,7)(10,13)(14,16)(21,35)(26,33)(29,34)(31,47)(39,42)(40,48)(43,46)(44,45)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 7185257);