Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| 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)(5,6,9,12)(7,13,14,10)(8,11,16,18)(15,19,20,17)(21,22,26,23)(24,29,32,27)(25,30,33,28)(31,34,36,35)(38,42)(39,45)(44,48)(47,51);(7,14)(10,13)(15,20)(17,19)(22,23)(25,33)(27,29)(31,36)(37,38,41,42)(39,43,45,46)(40,48,49,44)(47,52,51,50);(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35)(37,39)(38,43)(40,47)(41,45)(42,46)(44,50)(48,52)(49,51);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,17)(12,13)(16,20)(18,19)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35)(37,38)(39,46)(40,44)(41,42)(43,45)(47,52)(48,49)(50,51);(5,8,9,16)(6,11,12,18)(7,15,14,20)(10,17,13,19)(21,25,26,33)(22,28,23,30)(24,31,32,36)(27,34,29,35)(37,40)(38,44)(39,47)(41,49)(42,48)(43,50)(45,51)(46,52)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=h,d^2=gi,e^2=fg,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, 4408111);