Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,8)(7,10,11,12)(13,14,18,19)(15,16,22,20)(17,25,26,21)(23,28,27,24)(31,37)(33,41)(35,38)(36,40);(7,11)(10,12)(14,19)(15,22)(17,26)(24,28)(29,30,34,32)(31,38,37,35)(33,40,41,36)(39,42,43,44);(5,7,9,11)(6,10,8,12)(13,15,18,22)(14,20,19,16)(17,27,26,23)(21,24,25,28)(29,31,34,37)(30,35,32,38)(33,39,41,43)(36,42,40,44);(1,2)(3,4)(13,16,18,20)(14,15,19,22)(17,28,26,24)(21,27,25,23)(29,32,34,30)(31,35,37,38)(33,40,41,36)(39,42,43,44);(5,8,9,6)(7,12,11,10)(13,17,18,26)(14,21,19,25)(15,23,22,27)(16,24,20,28)(29,33)(30,36)(31,39)(32,40)(34,41)(35,42)(37,43)(38,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=h,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4059308);