Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,27,32)(23,33,35,45)(24,38,42,47)(25,31,26,30)(28,36,44,37)(29,40,48,41)(34,43,39,46)(49,51,52,50);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,26,36,27,35,25,37)(22,28,31,45,32,44,30,33)(24,39,41,51,42,34,40,50)(29,46,38,49,48,43,47,52);(7,13)(10,17)(14,15)(18,19)(21,24,25,40,27,42,26,41)(22,29,30,47,32,48,31,38)(23,34,37,51,35,39,36,50)(28,43,33,49,44,46,45,52);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(23,35)(28,44)(33,45)(34,39)(36,37)(43,46)(49,52)(50,51);(6,12)(10,17)(11,20)(18,19)(21,25,27,26)(22,30,32,31)(23,36,35,37)(24,40,42,41)(28,45,44,33)(29,47,48,38)(34,50,39,51)(43,52,46,49)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=hi,e^2=i,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^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6391852);