Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622979 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #60649 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,25)(12,18)(13,20)(14,39)(15,42)(16,43)(17,24)(19,30)(21,26)(22,41)(23,40)(27,53)(28,46)(29,45)(31,44)(32,55)(33,54)(34,47)(35,49)(36,65)(37,66)(38,67)(48,57)(50,56)(51,63)(52,64)(58,70)(59,69)(60,68)(61,72)(62,71);(2,5)(6,8)(9,11,26,33,16,29,41,34,17,30,15,28,40,32,14,12)(10,18,39,55,23,46,42,19,24,47,22,45,43,54,21,25)(13,27,56,62,38,59,63,60,35,57,37,58,64,61,36,31)(20,44,65,72,52,70,66,48,49,68,51,69,67,71,50,53);(1,3,4,7)(2,6,5,8)(9,12,14,32,40,28,15,30,17,34,41,29,16,33,26,11)(10,19,21,46,43,55,22,18,24,25,42,54,23,45,39,47)(13,31,36,61,64,58,37,57,35,60,63,59,38,62,56,27)(20,48,50,70,67,72,51,44,49,53,66,71,52,69,65,68);(9,13)(10,20)(11,27)(12,31)(14,36)(15,37)(16,38)(17,35)(18,44)(19,48)(21,50)(22,51)(23,52)(24,49)(25,53)(26,56)(28,58)(29,59)(30,57)(32,61)(33,62)(34,60)(39,65)(40,64)(41,63)(42,66)(43,67)(45,69)(46,70)(47,68)(54,71)(55,72);(13,35)(20,49)(27,57)(31,60)(36,63)(37,56)(38,64)(44,68)(48,53)(50,66)(51,65)(52,67)(58,62)(59,61)(69,72)(70,71)Polycyclic presentation:
2^9|b^2=gi,c^2=f,f^2=hi,g^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,h^a=hi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530375);