Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417682 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(2,5)(6,7)(9,10,16,22)(11,23,28,41)(12,29,33,44)(13,34,38,47)(14,21,15,20)(17,27,42,26)(18,31,45,32)(19,36,48,37)(24,51,55,39)(25,40,57,64)(30,43,63,69)(35,50,56,46)(49,53,58,54)(52,70,72,67)(59,61,68,62)(60,71,65,66);(1,2,4,5)(3,7,8,6)(9,11)(10,17)(12,24)(13,35)(14,27)(15,26)(16,28)(18,39)(19,46)(20,41)(21,23)(22,42)(25,36)(29,58)(30,60)(31,54)(32,53)(33,55)(34,64)(37,57)(38,56)(40,47)(43,67)(44,49)(45,51)(48,50)(52,61)(59,66)(62,72)(63,65)(68,71)(69,70);(9,12,16,33)(10,18,22,45)(11,24,28,55)(13,30,38,63)(14,32,15,31)(17,39,42,51)(19,43,48,69)(20,29,21,44)(23,49,41,58)(25,52,57,72)(26,54,27,53)(34,59,47,68)(35,65,56,60)(36,62,37,61)(40,66,64,71)(46,70,50,67);(1,3,4,8)(2,6,5,7)(9,13)(10,19)(11,25)(12,30)(14,37)(15,36)(16,38)(17,40)(18,43)(20,34)(21,47)(22,48)(23,50)(24,52)(26,56)(27,35)(28,57)(29,59)(31,62)(32,61)(33,63)(39,66)(41,46)(42,64)(44,68)(45,69)(49,67)(51,71)(53,65)(54,60)(55,72)(58,70)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336845);