Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623604 |
| 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 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1273 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)(6,8)(9,10)(11,22)(12,27)(13,19)(14,20)(15,34)(16,37)(17,25)(18,30)(21,36)(23,50)(24,42)(26,57)(28,61)(29,62)(31,40)(32,46)(33,60)(35,43)(38,47)(39,67)(41,64)(44,63)(45,65)(48,53)(49,55)(51,71)(52,59)(54,69)(56,66)(58,68)(70,72);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,23)(13,31)(14,25)(15,26)(16,24)(18,39)(19,42)(20,22)(21,41)(27,59)(28,54)(29,52)(30,53)(32,64)(33,56)(34,66)(35,58)(36,55)(37,40)(38,51)(43,70)(44,48)(45,68)(46,49)(47,69)(50,62)(57,60)(61,71)(63,67)(65,72);(9,12,14,29)(10,18,20,44)(11,23,25,52)(13,28,16,38)(15,35,33,65)(17,39,22,48)(19,43,37,45)(21,47,46,61)(24,51,31,54)(26,58,56,72)(27,60,62,34)(30,32,63,36)(40,68,42,70)(41,69,49,71)(50,66,59,57)(53,64,67,55);(2,5)(6,8)(9,13,14,16)(10,19,20,37)(11,24,25,31)(12,28,29,38)(15,32,33,36)(17,40,22,42)(18,43,44,45)(21,34,46,60)(23,51,52,54)(26,55,56,64)(27,61,62,47)(30,35,63,65)(39,68,48,70)(41,57,49,66)(50,71,59,69)(53,72,67,58);(1,3,4,7)(2,6,5,8)(11,25)(17,22)(23,52)(24,31)(26,56)(39,48)(40,42)(41,49)(50,59)(51,54)(53,67)(55,64)(57,66)(58,72)(68,70)(69,71)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531231);