Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394090 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,29,30,18)(14,19)(15,20)(21,35)(22,49,50,36)(23,37)(24,38)(25,39)(26,40,54,56)(27,41)(28,42)(31,57,58,43)(32,44)(33,45)(34,46)(47,61,69,70)(48,62)(51,67,59,63)(52,64)(53,65,71,72)(55,66,60,68);(2,5)(6,8)(9,11,15,24)(10,16,20,38)(12,25,28,52)(13,22,30,50)(14,32,34,23)(17,39,42,64)(18,36,29,49)(19,44,46,37)(21,33,48,27)(26,53,54,71)(31,59,58,51)(35,45,62,41)(40,65,56,72)(43,67,57,63)(47,60,69,55)(61,68,70,66);(9,12,15,28)(10,17,20,42)(11,21,24,48)(13,26,30,54)(14,33,34,27)(16,35,38,62)(18,40,29,56)(19,45,46,41)(22,47,50,69)(23,52,32,25)(31,60,58,55)(36,61,49,70)(37,64,44,39)(43,68,57,66)(51,71,59,53)(63,72,67,65);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,22)(12,26)(14,31)(15,30)(16,36)(17,40)(19,43)(20,29)(21,47)(23,51)(24,50)(25,53)(27,55)(28,54)(32,59)(33,60)(34,58)(35,61)(37,63)(38,49)(39,65)(41,66)(42,56)(44,67)(45,68)(46,57)(48,69)(52,71)(62,70)(64,72)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303237);