Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623326 |
| 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 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,22)(12,27)(13,20)(14,17)(15,34)(16,37)(18,26)(19,29)(21,36)(23,52)(24,51)(25,48)(28,46)(30,62)(31,44)(32,40)(33,49)(35,42)(38,47)(39,57)(41,67)(43,65)(45,64)(50,54)(53,71)(55,58)(56,70)(59,60)(61,66)(63,68)(69,72);(1,3,4,8)(2,5,6,7)(10,17)(18,39)(19,27)(20,37)(21,40)(22,48)(34,49)(41,58)(42,43)(44,46)(45,59)(47,51)(50,52)(62,66)(68,69)(70,71);(3,8)(5,7)(9,11,14,25)(10,18,17,39)(12,23,29,54)(13,24,16,38)(15,35,33,65)(19,41,27,58)(20,42,37,43)(21,47,40,51)(22,49,48,34)(26,32,57,36)(28,53,31,56)(30,63,61,72)(44,68,46,69)(45,70,59,71)(50,62,52,66)(55,64,67,60);(2,6)(5,7)(9,12)(10,19)(11,23)(13,31)(14,29)(15,30)(16,28)(17,27)(18,41)(20,46)(21,45)(22,50)(24,56)(25,54)(26,55)(32,64)(33,61)(34,66)(35,63)(36,60)(37,44)(38,53)(39,58)(40,59)(42,69)(43,68)(47,70)(48,52)(49,62)(51,71)(57,67)(65,72);(9,13,14,16)(10,20,17,37)(11,24,25,38)(12,28,29,31)(15,32,33,36)(18,42,39,43)(19,44,27,46)(21,34,40,49)(22,51,48,47)(23,53,54,56)(26,35,57,65)(30,60,61,64)(41,68,58,69)(45,62,59,66)(50,70,52,71)(55,72,67,63)Polycyclic presentation:
2^9|b^2=fi,c^2=i,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=cfi,g^c=gh,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530582);