Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368558 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,8)(5,6)(9,21)(10,22)(11,19)(13,18)(15,26)(16,20)(23,31)(24,25)(27,32)(28,29)(33,34,38,42)(35,44,47,56)(36,41,50,58)(37,53,54,43)(39,49,57,48)(40,51,59,52)(45,62,64,55)(46,61,63,60);(1,2)(3,9)(4,15)(5,8)(6,19)(7,11)(10,16)(12,27)(13,24)(14,25)(17,22)(18,21)(20,26)(23,29)(28,32)(30,31)(33,35)(34,39)(36,45)(37,49)(38,47)(40,55)(41,60)(42,57)(43,56)(44,53)(46,51)(48,54)(50,64)(52,63)(58,61)(59,62);(1,3,7,14)(2,9,11,25)(4,12,17,30)(5,18,6,13)(8,21,19,24)(10,23,26,32)(15,27,22,31)(16,29,20,28)(33,36)(34,40)(35,45)(37,51)(38,50)(39,55)(41,53)(42,59)(43,58)(44,60)(46,49)(47,64)(48,63)(52,54)(56,61)(57,62);(1,4)(2,10)(3,12)(5,16)(6,20)(7,17)(8,22)(9,23)(11,26)(13,28)(14,30)(15,19)(18,29)(21,31)(24,27)(25,32)(33,36)(34,41)(35,46)(37,52)(38,50)(39,55)(40,53)(42,58)(43,59)(44,61)(45,49)(47,63)(48,64)(51,54)(56,60)(57,62);(1,5,7,6)(2,8,11,19)(3,13,14,18)(4,16,17,20)(9,24,25,21)(10,22,26,15)(12,28,30,29)(23,27,32,31)(34,42)(35,47)(36,50)(43,53)(48,49)(51,52)(55,62)(60,61)Polycyclic presentation:
2^9|a^2=i,c^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72306669, #72306694, #72306699, … |
| maximal normal subgroup of | #72306669, #72306694, #72306699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6275889);