Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234321 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1087 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,38,40)(35,45,48,56)(36,42,50,54)(37,44,39,43)(41,47,53,49)(46,59,60,64)(51,58,52,57)(55,62,63,61);(1,3,5,10)(2,6,8,9)(4,11,13,16)(7,14,12,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(34,40)(36,50)(41,53)(43,44)(45,56)(46,60)(51,52)(61,62);(19,25)(20,27)(22,28)(23,30)(33,35,37,47,38,48,39,49)(34,41,43,56,40,53,44,45)(36,46,52,62,50,60,51,61)(42,55,58,64,54,63,57,59);(3,10)(6,9)(11,16)(14,15)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,36)(34,42)(35,46)(37,52)(38,50)(39,51)(40,54)(41,55)(43,58)(44,57)(45,59)(47,62)(48,60)(49,61)(53,63)(56,64);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(19,25)(22,28)(26,31)(29,32)(33,37,38,39)(34,43,40,44)(35,47,48,49)(36,51,50,52)(41,56,53,45)(42,57,54,58)(46,61,60,62)(55,59,63,64)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=gi,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7200348);