Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470161 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,27,32)(23,28,35,43)(24,36,38,46)(25,31,26,30)(29,40,44,37)(33,47,49,51)(34,45,39,42)(41,48,52,50);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,28)(24,33)(25,39)(26,34)(27,35)(29,41)(30,45)(31,42)(32,43)(36,47)(37,50)(38,49)(40,48)(44,52)(46,51);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24)(22,29)(23,33)(25,37)(26,40)(27,38)(28,41)(30,36)(31,46)(32,44)(34,48)(35,49)(39,50)(42,51)(43,52)(45,47);(7,14)(10,18)(13,15)(17,19)(21,25,27,26)(22,30,32,31)(23,34,35,39)(24,37,38,40)(28,42,43,45)(29,36,44,46)(33,48,49,50)(41,51,52,47);(6,12)(10,18)(11,20)(17,19)(23,35)(28,43)(33,49)(34,39)(41,52)(42,45)(47,51)(48,50)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6384855);