Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 | |
1778 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)(5,6,8,10)(7,9,11,12)(13,14,18,19)(15,16,22,20)(17,25,27,21)(23,26,28,24)(29,30)(31,34)(32,40)(33,37)(35,39)(36,41)(38,43)(42,44);(7,11)(9,12)(14,19)(15,22)(21,25)(23,28)(29,31,33,39)(30,34,37,35)(32,38,41,44)(36,42,40,43);(5,7)(6,9)(8,11)(10,12)(13,15,18,22)(14,20,19,16)(17,23,27,28)(21,26,25,24)(29,31)(30,35)(32,38)(33,39)(34,37)(36,43)(40,42)(41,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,20)(14,15,19,22)(17,26,27,24)(21,28,25,23)(29,30)(31,35)(32,36)(33,37)(34,39)(38,43)(40,41)(42,44);(13,17,18,27)(14,21,19,25)(15,23,22,28)(16,24,20,26)(29,32,33,41)(30,36,37,40)(31,38,39,44)(34,42,35,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2643624);