Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1337 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,33)(27,31,38,45)(28,32,39,47)(29,34,41,48)(35,42,37,44)(36,43,51,55)(40,46,52,56)(49,53,50,54);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,31)(28,36)(29,35)(30,38)(32,43)(33,45)(34,42)(37,41)(39,51)(40,49)(44,48)(46,53)(47,55)(50,52)(54,56);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(27,35)(29,41)(31,42)(34,48)(36,49)(37,38)(40,52)(43,53)(44,45)(46,56)(50,51)(54,55);(10,16)(14,17)(15,22)(21,23)(25,28)(26,32)(27,36)(29,40)(30,39)(31,43)(33,47)(34,46)(35,49)(37,50)(38,51)(41,52)(42,53)(44,54)(45,55)(48,56);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(26,33)(27,35)(28,39)(29,41)(31,44)(36,50)(37,38)(42,45)(43,53)(46,56)(49,51)(54,55)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fg,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6629136);