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 | |
1339 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)(27,32,40,49)(28,41,44,33)(29,34,31,36)(37,53,55,46)(38,47,39,48)(42,56,54,50)(43,52,45,51);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,32)(28,42)(29,38)(30,40)(31,39)(33,50)(34,47)(35,49)(36,48)(37,43)(41,56)(44,54)(45,55)(46,51)(52,53);(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)(29,43)(30,44)(31,45)(32,46)(34,51)(35,41)(36,52)(38,54)(39,42)(40,55)(47,56)(48,50)(49,53);(10,16)(14,17)(15,22)(21,23)(25,29,30,31)(26,34,35,36)(27,38,40,39)(28,43,44,45)(32,47,49,48)(33,51,41,52)(37,54,55,42)(46,56,53,50);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(27,39)(28,44)(29,31)(32,48)(33,41)(34,36)(37,54)(38,40)(42,55)(46,56)(47,49)(50,53)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fgi,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6629079);