Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 52 points:
(2,4)(5,6,10,11)(7,15,18,12)(8,13,9,14)(16,20,17,19)(25,37)(30,39)(33,44)(35,48)(41,49)(42,47)(46,51)(50,52);(1,2)(3,4)(6,11)(12,15)(13,14)(19,20)(22,27)(23,31)(29,34)(30,39)(32,43)(33,44)(42,47)(46,51);(5,6)(7,12)(8,13)(9,14)(10,11)(15,18)(16,19)(17,20)(21,22)(23,28)(24,34)(25,30)(26,27)(29,36)(31,38)(32,45)(33,41)(35,47)(37,39)(40,43)(42,48)(44,49)(46,52)(50,51);(5,7)(6,12)(8,16)(9,17)(10,18)(11,15)(13,19)(14,20)(21,23,26,31)(22,28,27,38)(24,32,36,43)(25,33,37,44)(29,40,34,45)(30,41,39,49)(35,46,48,51)(42,50,47,52);(5,8,10,9)(6,13,11,14)(7,16,18,17)(12,19,15,20)(21,24,26,36)(22,29,27,34)(23,32,31,43)(25,35,37,48)(28,40,38,45)(30,42,39,47)(33,46,44,51)(41,50,49,52);(5,9,10,8)(6,14,11,13)(7,17,18,16)(12,20,15,19)(21,25)(22,30)(23,33)(24,35)(26,37)(27,39)(28,41)(29,42)(31,44)(32,46)(34,47)(36,48)(38,49)(40,50)(43,51)(45,52)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551538);