Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631116 |
| 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 | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,33)(28,42)(29,35)(30,36)(31,46)(32,37)(34,43)(38,48)(39,49)(40,55)(41,50)(44,52)(45,51)(47,53)(54,56);(3,7)(5,8)(10,14)(15,20)(16,21)(22,24)(25,27)(26,33)(28,38)(29,44)(30,39)(31,40)(32,41)(34,47)(35,52)(36,49)(37,50)(42,48)(43,53)(45,54)(46,55)(51,56);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,32,29)(26,34,37,51)(27,38,41,44)(30,43,31,45)(33,47,50,56)(35,46,42,36)(39,53,40,54)(48,49,52,55);(2,6)(3,7)(11,17)(15,20)(18,23)(22,24)(25,29,32,28)(26,35,37,42)(27,38,41,44)(30,43,31,45)(33,48,50,52)(34,46,51,36)(39,54,40,53)(47,49,56,55);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,30,32,31)(26,36,37,46)(27,39,41,40)(28,43,29,45)(33,49,50,55)(34,35,51,42)(38,53,44,54)(47,52,56,48)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,d^c=dfh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538645);