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 | |
1335 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)(25,26,29,32)(27,35,39,31)(28,40,41,33)(30,34,43,46)(36,44,38,48)(37,45,51,49)(42,53,54,47)(50,56,52,55);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,29,39)(26,31,32,35)(28,37,41,51)(30,36,43,38)(33,45,40,49)(34,48,46,44)(42,50,54,52)(47,56,53,55);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(27,36)(30,43)(31,44)(33,40)(35,48)(37,50)(38,39)(42,54)(45,55)(49,56)(51,52);(10,16)(14,17)(15,22)(21,23)(25,28)(26,33)(27,37)(29,41)(30,42)(31,45)(32,40)(34,47)(35,49)(36,50)(38,52)(39,51)(43,54)(44,55)(46,53)(48,56);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(27,38)(28,41)(30,43)(31,44)(33,40)(34,46)(35,48)(36,39)(37,50)(45,56)(49,55)(51,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fg,e^2=gi,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, 6629009);