Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #30430 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1716 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)(3,4)(5,7)(6,8)(9,10)(11,22)(12,27)(13,33)(14,20)(15,37)(16,21)(17,25)(18,31)(19,38)(23,44)(24,54)(26,42)(28,51)(29,60)(30,50)(32,46)(34,66)(35,49)(36,48)(39,43)(40,57)(41,55)(45,62)(47,56)(52,68)(53,69)(58,64)(59,70)(61,71)(63,65)(67,72);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,28)(13,34)(14,25)(15,26)(16,39)(18,44)(19,47)(20,22)(21,42)(23,32)(24,35)(27,40)(29,61)(30,57)(31,53)(33,54)(36,56)(37,43)(38,55)(41,48)(45,71)(46,69)(49,66)(50,51)(52,64)(58,72)(59,62)(60,70)(63,67)(65,68);(9,12,14,30)(10,18,20,46)(11,23,25,53)(13,29,35,62)(15,32,16,31)(17,40,22,51)(19,45,48,60)(21,50,37,27)(24,52,34,67)(26,57,39,28)(33,65,49,58)(36,64,38,63)(41,68,47,72)(42,44,43,69)(54,71,66,70)(55,59,56,61);(2,7)(3,8)(9,13)(10,19)(11,24)(12,29)(14,35)(15,38)(16,36)(17,41)(18,45)(20,48)(21,33)(22,47)(23,52)(25,34)(26,56)(27,58)(28,59)(30,62)(31,64)(32,63)(37,49)(39,55)(40,68)(42,66)(43,54)(44,70)(46,60)(50,65)(51,72)(53,67)(57,61)(69,71);(1,4,5,6)(2,3,7,8)(12,30)(18,46)(23,53)(27,50)(28,57)(29,62)(31,32)(40,51)(44,69)(45,60)(52,67)(58,65)(59,61)(63,64)(68,72)(70,71)Polycyclic presentation:
2^9|c^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409752);