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)(25,26,30,35)(27,36,39,32)(28,33,42,51)(29,43,31,34)(37,53,55,46)(38,47,44,48)(40,56,54,49)(41,52,45,50);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,30,39)(26,32,35,36)(28,40,42,54)(29,38,31,44)(33,49,51,56)(34,47,43,48)(37,45,55,41)(46,50,53,52);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,33)(27,37)(29,41)(30,42)(31,45)(32,46)(34,50)(35,51)(36,53)(38,54)(39,55)(40,44)(43,52)(47,49)(48,56);(10,16)(14,17)(15,22)(21,23)(25,29,30,31)(26,34,35,43)(27,38,39,44)(28,41,42,45)(32,47,36,48)(33,50,51,52)(37,54,55,40)(46,49,53,56);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(27,38)(29,31)(32,48)(34,43)(36,47)(37,54)(39,44)(40,55)(41,45)(46,56)(49,53)(50,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fgi,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6629014);