Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,13)(8,16)(9,10)(11,15)(12,17)(14,18)(19,20)(21,22,26,24)(23,29,30,27)(25,32,33,28)(31,34,36,35)(39,45)(40,47)(42,48)(43,50);(6,10)(8,17)(11,13)(14,20)(22,24)(23,30)(25,33)(34,35)(37,38,41,44)(39,42,45,48)(40,43,47,50)(46,49,51,52);(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,23,26,30)(22,27,24,29)(25,31,33,36)(28,34,32,35)(37,38)(39,42)(40,43)(41,44)(45,48)(46,49)(47,50)(51,52);(1,2)(3,4)(5,7,9,15)(6,11,10,13)(8,14,17,20)(12,18,16,19)(21,24,26,22)(23,27,30,29)(25,32,33,28)(31,34,36,35)(37,39,41,45)(38,42,44,48)(40,46,47,51)(43,49,50,52);(5,8,9,17)(6,12,10,16)(7,14,15,20)(11,18,13,19)(21,25,26,33)(22,28,24,32)(23,31,30,36)(27,34,29,35)(37,40)(38,43)(39,46)(41,47)(42,49)(44,50)(45,51)(48,52)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4375786);