Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,7)(4,8)(5,6)(9,10,14,17)(11,19,21,15)(12,16,13,18)(20,24,22,23)(25,26)(27,35)(28,39)(29,33)(30,34)(31,38)(32,41)(36,46)(37,51)(40,48)(42,52)(43,50)(44,47)(45,49)(53,54)(55,56);(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(25,27)(26,31)(28,40)(29,38)(30,44)(32,48)(33,35)(34,51)(36,42)(37,43)(39,56)(41,55)(45,54)(46,49)(47,50)(52,53);(2,6)(7,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28)(26,32)(27,36)(29,41)(30,45)(31,46)(33,39)(34,52)(35,53)(37,55)(38,54)(40,44)(42,43)(47,56)(48,51)(49,50);(1,3,5,4)(2,7,6,8)(10,17)(15,19)(16,18)(23,24)(26,33)(28,41)(31,35)(34,50)(36,54)(40,55)(42,45)(47,51);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(27,37)(30,43)(31,47)(34,50)(35,51)(36,55)(38,44)(40,54)(42,45)(46,56)(48,53)(49,52)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fi,d^2=gi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6648190);