Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623308 |
| 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 | #60643 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1620 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)(3,7)(4,6)(5,8)(9,10,16,22)(11,17,24,39)(12,28,31,60)(13,21,15,19)(14,33,36,46)(18,43,45,30)(20,32,48,35)(23,50,51,70)(25,42,27,40)(26,56,54,67)(29,61,37,47)(34,49,62,44)(38,63,64,53)(41,55,66,58)(52,71,59,68)(57,69,72,65);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,23)(13,25)(14,26)(15,27)(16,24)(18,38)(19,40)(20,41)(21,42)(22,39)(28,50)(29,52)(30,53)(31,51)(32,55)(33,56)(34,57)(35,58)(36,54)(37,59)(43,63)(44,65)(45,64)(46,67)(47,68)(48,66)(49,69)(60,70)(61,71)(62,72);(3,8)(5,7)(9,12)(10,18)(11,23)(13,29)(14,34)(15,37)(16,31)(17,38)(19,44)(20,47)(21,49)(22,45)(24,51)(25,52)(26,57)(27,59)(28,33)(30,35)(32,43)(36,62)(39,64)(40,65)(41,68)(42,69)(46,60)(48,61)(50,56)(53,58)(54,72)(55,63)(66,71)(67,70);(2,6)(5,7)(11,24)(17,39)(23,51)(25,27)(26,54)(38,64)(40,42)(41,66)(50,70)(52,59)(53,63)(55,58)(56,67)(57,72)(65,69)(68,71);(9,13,16,15)(10,19,22,21)(11,25,24,27)(12,29,31,37)(14,32,36,35)(17,40,39,42)(18,44,45,49)(20,46,48,33)(23,52,51,59)(26,55,54,58)(28,47,60,61)(30,34,43,62)(38,65,64,69)(41,67,66,56)(50,68,70,71)(53,57,63,72)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gh,h^a=hi,c^b=cf,d^b=di,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530560);