Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395628 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14725 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15)(4,10)(5,26,28,38)(6,13)(7,12)(8,14)(9,18)(11,27,39,29)(16,41)(17,31,46,58)(19,32)(20,40)(21,30)(22,50,52,55)(23,36)(24,35)(25,37)(33,44)(34,53,60,51)(42,45,54,47)(43,62,63,56)(48,57)(49,61,64,59);(1,3,6,18,8,20,7,19)(2,9,12,15,14,32,13,40)(4,21,23,44,25,48,24,16)(5,17,27,45,28,46,29,47)(10,33,35,30,37,41,36,57)(11,31,38,42,39,58,26,54)(22,49,51,63,52,64,53,43)(34,59,50,56,60,61,55,62);(1,4,8,25)(2,10,14,37)(3,16,20,44)(5,22,28,52)(6,23,7,24)(9,30,32,57)(11,34,39,60)(12,35,13,36)(15,41,40,33)(17,43,46,63)(18,21,19,48)(26,55,38,50)(27,51,29,53)(31,56,58,62)(42,61,54,59)(45,49,47,64);(1,5)(2,11)(3,17)(4,22)(6,27)(7,29)(8,28)(9,31)(10,34)(12,38)(13,26)(14,39)(15,42)(16,43)(18,45)(19,47)(20,46)(21,49)(23,51)(24,53)(25,52)(30,56)(32,58)(33,59)(35,50)(36,55)(37,60)(40,54)(41,61)(44,63)(48,64)(57,62)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72287080, #72287270, #72287312, … |
| maximal normal subgroup of | #72287080, #72287270, #72287312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302825);