Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394685 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14718 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2083 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)(4,20)(5,26)(6,13)(7,12)(8,14)(9,18)(10,24)(11,27)(16,35)(17,32)(19,33)(21,49)(22,52)(23,37)(25,38)(28,39)(29,34)(30,40)(31,51)(36,55)(41,44)(42,46)(43,62)(45,56)(47,59)(48,53)(50,61)(54,57)(58,63)(60,64);(1,3)(2,9)(4,21)(5,17)(6,29)(7,18)(8,19)(10,35)(11,32)(12,33)(13,15)(14,34)(16,23)(20,41)(22,50)(24,51)(25,44)(26,56)(27,46)(28,47)(30,45)(31,38)(36,60)(37,49)(39,42)(40,59)(43,53)(48,61)(52,58)(54,64)(55,63)(57,62);(1,4,6,23,8,24,7,25)(2,10,12,37,14,20,13,38)(3,16,18,21,19,44,29,51)(5,22,27,53,28,54,30,55)(9,31,15,41,34,49,33,35)(11,36,26,57,40,48,39,52)(17,43,45,50,47,63,46,64)(32,58,42,61,59,62,56,60);(1,5,8,28)(2,11,14,40)(3,17,19,47)(4,22,24,54)(6,27,7,30)(9,32,34,59)(10,36,20,48)(12,26,13,39)(15,42,33,56)(16,43,44,63)(18,45,29,46)(21,50,51,64)(23,53,25,55)(31,58,49,62)(35,60,41,61)(37,57,38,52)Polycyclic presentation:
2^9|c^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72287280, #72287292, #72287334, … |
| maximal normal subgroup of | #72287280, #72287292, #72287334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301882);