Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1959 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,35)(24,39)(25,31)(26,32)(27,33)(28,34)(29,37)(30,38)(36,49)(40,52)(41,46)(42,47)(43,48)(44,50)(45,51);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,27,24,28,38,26,37)(22,29,32,30,34,39,33,35)(25,36,41,40,43,51,42,50)(31,44,47,45,48,52,46,49);(7,13)(8,15)(10,16)(11,18)(21,24,27,38,28,37,26,23)(22,30,32,39,34,35,33,29)(25,40,41,51,43,50,42,36)(31,45,47,52,48,49,46,44);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,25,28,43)(22,31,34,48)(23,36,38,51)(24,40,37,50)(26,42,27,41)(29,44,39,52)(30,45,35,49)(32,47,33,46);(7,13)(10,16)(14,19)(17,20)(21,26,28,27)(22,32,34,33)(23,37,38,24)(25,41,43,42)(29,30,39,35)(31,46,48,47)(36,40,51,50)(44,49,52,45)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6434270);