Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| 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×#3212 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2041 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)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22,27,31)(23,33,37,45)(24,38,41,47)(25,30,26,32)(28,35,44,36)(29,40,48,42)(34,43,39,46)(49,51,52,50);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,25,36,27,37,26,35)(22,28,32,33,31,44,30,45)(24,39,40,50,41,34,42,51)(29,46,47,52,48,43,38,49);(6,11)(7,13)(12,15)(14,19)(21,24,25,40,27,41,26,42)(22,29,32,47,31,48,30,38)(23,34,36,51,37,39,35,50)(28,43,33,49,44,46,45,52);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,31,32)(23,35,37,36)(24,40,41,42)(28,33,44,45)(29,38,48,47)(34,50,39,51)(43,49,46,52);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,31)(24,41)(28,44)(30,32)(33,45)(34,39)(40,42)(50,51)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gh,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406068);