Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3307 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1863 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,26,21)(16,22,30,37)(17,23,18,20)(27,39,40,36)(28,35,31,33)(29,38,32,34)(41,44,42,43);(5,6)(7,11)(8,10)(9,12)(14,20)(15,26)(16,29)(17,18)(21,35)(22,37)(23,24)(25,33)(27,41)(30,32)(40,42)(43,44);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,27)(17,31)(18,28)(19,26)(20,33)(22,36)(23,35)(24,25)(29,42)(30,40)(32,41)(34,43)(37,39)(38,44);(7,12)(9,11)(13,16,19,30)(14,22,24,37)(15,27,26,40)(17,29,18,32)(20,34,23,38)(21,36,25,39)(28,41,31,42)(33,43,35,44);(6,10)(9,11)(13,17,19,18)(14,20,24,23)(15,28,26,31)(16,29,30,32)(21,35,25,33)(22,34,37,38)(27,41,40,42)(36,44,39,43)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6969938);