Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623311 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(3,7)(9,10)(11,22)(12,27)(13,20)(14,19)(15,35)(16,38)(17,25)(18,31)(21,37)(23,51)(24,43)(26,57)(28,61)(29,60)(30,59)(32,41)(33,48)(34,65)(36,46)(39,49)(40,62)(42,64)(44,53)(45,63)(47,66)(50,55)(52,72)(54,70)(56,67)(58,69)(68,71);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,28)(13,32)(14,25)(15,26)(16,24)(18,44)(19,22)(20,43)(21,42)(23,30)(27,51)(29,52)(31,62)(33,64)(34,56)(35,67)(36,68)(37,55)(38,41)(39,54)(40,45)(46,69)(47,71)(48,50)(49,72)(53,63)(57,65)(58,66)(59,61)(60,70);(2,5)(6,8)(9,12)(10,18)(11,23)(13,29)(14,30)(15,36)(16,39)(17,40)(19,45)(20,46)(21,49)(22,44)(24,52)(25,28)(26,58)(27,35)(31,37)(32,54)(33,63)(34,66)(38,47)(41,69)(42,70)(43,71)(48,60)(50,72)(51,57)(53,55)(56,68)(59,65)(61,67)(62,64);(1,3,4,7)(2,6,5,8)(10,19)(17,22)(18,45)(20,38)(21,48)(27,59)(35,65)(40,44)(41,43)(42,50)(46,47)(49,60)(51,61)(57,67)(69,71)(70,72);(9,13,14,16)(10,20,19,38)(11,24,25,32)(12,29,30,39)(15,33,34,37)(17,41,22,43)(18,46,45,47)(21,35,48,65)(23,52,28,54)(26,55,56,64)(27,60,59,49)(31,36,63,66)(40,69,44,71)(42,57,50,67)(51,72,61,70)(53,68,62,58)Polycyclic presentation:
2^9|d^2=fi,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cf,e^b=ei,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530683);