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 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,17)(12,27)(13,19)(14,32)(15,36)(16,21)(18,29)(20,34)(22,47)(23,39)(24,40)(25,53)(26,56)(28,58)(30,59)(31,46)(33,42)(35,61)(37,45)(38,50)(41,55)(43,62)(44,60)(48,68)(49,69)(51,66)(52,71)(54,63)(57,67)(64,70)(65,72);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,22)(13,23)(14,25)(15,26)(16,24)(18,38)(19,39)(20,41)(21,40)(27,47)(28,48)(29,50)(30,49)(31,51)(32,53)(33,54)(34,55)(35,52)(36,56)(37,57)(42,63)(43,65)(44,64)(45,67)(46,66)(58,68)(59,69)(60,70)(61,71)(62,72);(1,4,5,2)(3,7,8,6)(9,12)(10,18)(11,22)(13,28)(14,33)(15,37)(16,30)(17,38)(19,42)(20,45)(21,44)(23,48)(24,49)(25,54)(26,57)(27,32)(29,34)(31,60)(35,62)(36,43)(39,63)(40,64)(41,67)(46,58)(47,53)(50,55)(51,70)(52,72)(56,65)(59,61)(66,68)(69,71);(3,8)(6,7)(9,13,16,15)(10,19,21,36)(11,23,24,26)(12,28,30,37)(14,31,35,34)(17,39,40,56)(18,42,44,43)(20,32,46,61)(22,48,49,57)(25,51,52,55)(27,58,59,45)(29,33,60,62)(38,63,64,65)(41,53,66,71)(47,68,69,67)(50,54,70,72);(11,24)(17,40)(22,49)(23,26)(25,52)(38,64)(39,56)(41,66)(47,69)(48,57)(50,70)(51,55)(53,71)(54,72)(63,65)(67,68)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=i,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, 7531015);