Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26)(27,36)(28,32)(29,33)(30,35)(31,41)(34,38)(37,51)(39,53)(40,46)(42,48)(43,56)(44,50)(45,52)(47,54)(49,55);(1,2)(3,7)(4,6)(5,8)(9,11,14,20)(10,15,17,18)(12,19,13,21)(16,23,22,24)(27,37)(28,42)(31,46)(32,48)(36,51)(40,41)(44,45)(50,52);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,27,38,40,30,37,29,41)(26,31,33,51,35,46,34,36)(28,43,44,54,42,39,45,55)(32,49,52,53,48,47,50,56);(2,6)(3,8)(11,20)(15,18)(19,21)(23,24)(25,28,30,42)(26,32,35,48)(27,39,37,43)(29,45,38,44)(31,47,46,49)(33,52,34,50)(36,53,51,56)(40,55,41,54);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(25,29,30,38)(26,33,35,34)(27,40,37,41)(28,44,42,45)(31,36,46,51)(32,50,48,52)(39,54,43,55)(47,56,49,53)Polycyclic presentation:
2^9|b^2=f,c^2=fgi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7242103);