Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456656 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16284 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1368 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)(3,8,15,30)(4,9)(5,10)(6,23)(7,11)(12,27)(13,42,44,28)(14,29,46,55)(16,39,24,31)(17,32,48,57)(18,36)(19,40)(20,34)(21,41)(22,37)(25,35)(26,38)(33,52)(43,54,45,63)(47,58,49,56)(50,62,64,59)(51,60)(53,61);(1,3)(2,8,11,30)(4,17)(5,14)(6,24)(7,15)(9,32,34,57)(10,29,37,55)(12,39,23,31)(13,19)(16,27)(18,50)(20,48)(21,49)(22,46)(25,44)(26,47)(28,40,42,35)(33,59,36,62)(38,58,41,56)(43,53)(45,51)(52,64)(54,60,63,61);(1,4)(2,9)(3,13)(5,18)(6,25)(7,20)(8,28)(10,36)(11,34)(12,40)(14,45)(15,44)(16,48)(17,24)(19,27)(21,53)(22,52)(23,35)(26,51)(29,54)(30,42)(31,57)(32,39)(33,37)(38,60)(41,61)(43,46)(47,64)(49,50)(55,63)(56,59)(58,62);(1,5,7,22)(2,10,11,37)(3,14,15,46)(4,18,20,52)(6,26,27,21)(8,29,30,55)(9,33,34,36)(12,41,23,38)(13,43,44,45)(16,49,24,47)(17,50,48,64)(19,53,25,51)(28,54,42,63)(31,58,39,56)(32,59,57,62)(35,61,40,60)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72285845, #72286090, #72286126, … |
| maximal normal subgroup of | #72285845, #72286090, #72286126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363853);