Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470161 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2884 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)(21,22,26,29)(23,27,32,42)(24,34,37,46)(25,38,40,30)(28,39,45,36)(31,47,35,52)(33,50,49,43)(41,51,44,48);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23)(22,27)(24,35)(25,33)(26,32)(28,44)(29,42)(30,43)(31,37)(34,52)(36,51)(38,50)(39,48)(40,49)(41,45)(46,47);(7,13)(10,17)(14,15)(18,19)(21,24,26,37)(22,28,29,45)(23,31,32,35)(25,39,40,36)(27,41,42,44)(30,46,38,34)(33,51,49,48)(43,52,50,47);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(22,29)(27,42)(28,45)(30,38)(34,46)(41,44)(43,50)(47,52);(6,12)(10,17)(11,20)(18,19)(22,29)(23,32)(28,45)(30,38)(31,35)(33,49)(34,46)(48,51)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=hi,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, 6388633);