Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #24425 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1827 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)(13,14,20,24,19,26,18,25)(15,27,29,21,16,22,30,37)(17,23,33,42,35,41,34,40)(28,39,43,36,31,38,32,44);(5,6)(7,11)(8,10)(9,12)(13,15,19,16)(14,21,26,37)(17,32,35,43)(18,29,20,30)(22,24,27,25)(23,39,41,38)(28,33,31,34)(36,42,44,40);(2,4)(5,7,8,12)(6,9,10,11)(13,16,19,15)(14,22,26,27)(17,31,35,28)(18,29,20,30)(21,24,37,25)(23,38,41,39)(32,34,43,33)(36,42,44,40);(7,12)(9,11)(13,17,20,33,19,35,18,34)(14,23,24,42,26,41,25,40)(15,28,29,43,16,31,30,32)(21,36,22,38,37,44,27,39);(6,10)(9,11)(13,18,19,20)(14,24,26,25)(15,29,16,30)(17,33,35,34)(21,27,37,22)(23,40,41,42)(28,32,31,43)(36,38,44,39)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=hi,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972138);