Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61086 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(3,7)(6,8)(9,10,16,22,15,20,14,21)(11,23,28,17,26,52,27,42)(12,29,38,46,32,45,33,18)(13,19,39,51,37,49,36,50)(24,43,56,53,30,40,55,60)(25,47,59,41,58,71,34,64)(31,61,67,48,35,65,63,44)(54,69,62,72,57,68,66,70);(2,5)(3,7)(9,11)(10,17)(12,30)(13,34)(14,28)(15,26)(16,27)(18,43)(19,47)(20,42)(21,52)(22,23)(24,32)(25,36)(29,60)(31,62)(33,55)(35,66)(37,59)(38,56)(39,58)(40,46)(41,50)(44,70)(45,53)(48,72)(49,71)(51,64)(54,67)(57,63)(61,68)(65,69);(1,2)(3,6)(4,5)(7,8)(9,12,16,38,15,32,14,33)(10,18,21,45,20,46,22,29)(11,24,27,55,26,30,28,56)(13,35,39,63,37,31,36,67)(17,40,52,60,42,43,23,53)(19,48,50,61,49,44,51,65)(25,57,34,62,58,54,59,66)(41,69,71,72,64,68,47,70);(2,5)(6,8)(9,13,16,39,15,37,14,36)(10,19,22,51,20,49,21,50)(11,25,28,59,26,58,27,34)(12,31,38,67,32,35,33,63)(17,41,52,71,42,64,23,47)(18,44,29,61,46,48,45,65)(24,54,56,62,30,57,55,66)(40,68,60,70,43,69,53,72);(1,3,4,7)(2,6,5,8)(10,20)(11,26)(12,32)(19,49)(21,22)(25,58)(27,28)(31,35)(33,38)(34,59)(40,43)(50,51)(53,60)(63,67)(68,69)(70,72)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,d^2=fi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7063408);