Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26)(29,30)(31,37)(32,34)(33,40)(35,39)(36,42)(38,43)(41,44);(6,9)(10,12)(14,15)(19,21)(20,22)(26,27)(30,32)(35,37)(36,40)(43,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,21,24,19)(17,20,25,22)(23,26,28,27)(29,31)(30,35)(32,37)(33,41)(34,39)(36,44)(38,42)(40,43);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28)(29,32,34,30)(31,37,39,35)(33,40,42,36)(38,44,41,43);(5,7)(6,10)(8,11)(9,12)(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)(29,33)(30,36)(31,38)(32,40)(34,42)(35,43)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=fg,c^2=fg,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4574097);