Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1610 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)(15,16)(17,25)(18,19)(20,22)(21,28)(23,27)(24,26)(29,30)(31,38)(32,36)(33,35)(34,37)(39,44)(40,43)(41,42);(7,11)(10,12)(14,19)(15,22)(21,25)(23,26)(29,31,35,32)(30,36,33,38)(34,39,42,40)(37,43,41,44);(5,7,9,11)(6,10,8,12)(13,15)(14,20)(16,19)(17,26)(18,22)(21,24)(23,28)(25,27)(29,32,35,31)(30,36,33,38)(34,39,42,40)(37,44,41,43);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,16,18,20)(14,15,19,22)(17,27,28,24)(21,26,25,23)(29,33,35,30)(31,36,32,38)(34,37,42,41)(39,44,40,43);(5,8,9,6)(7,12,11,10)(13,17,18,28)(14,21,19,25)(15,23,22,26)(16,24,20,27)(29,34,35,42)(30,37,33,41)(31,39,32,40)(36,43,38,44)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fh,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3270503);