Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2875 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)(21,22,27,33)(23,34,38,44)(24,30,41,46)(25,31,26,32)(28,36,45,37)(29,40,47,39)(35,49,52,42)(43,51,48,50);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,27,38)(22,28,33,45)(24,35,41,52)(25,36,26,37)(29,42,47,49)(30,43,46,48)(31,34,32,44)(39,50,40,51);(7,13)(10,17)(14,15)(18,19)(21,24)(22,29)(23,35)(25,39)(26,40)(27,41)(28,42)(30,32)(31,46)(33,47)(34,48)(36,50)(37,51)(38,52)(43,44)(45,49);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(21,24)(22,30)(23,35)(25,40)(26,39)(27,41)(28,43)(29,32)(31,47)(33,46)(34,49)(36,51)(37,50)(38,52)(42,44)(45,48);(6,12)(10,17)(11,20)(18,19)(21,25,27,26)(22,31,33,32)(23,36,38,37)(24,39,41,40)(28,34,45,44)(29,46,47,30)(35,50,52,51)(42,48,49,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=h,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,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388933);