Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #45094 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1836 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)(15,27,17,21,30,39,29,23)(16,31,34,41,36,22,35,42)(28,44,37,40,32,43,33,38);(5,6)(7,11)(8,10)(9,12)(13,15,19,30)(14,21,26,23)(16,32,36,28)(17,20,29,18)(22,40,31,38)(24,27,25,39)(33,34,37,35)(41,43,42,44);(2,4)(5,7,8,12)(6,9,10,11)(13,16,19,36)(14,22,26,31)(15,28,30,32)(17,37,29,33)(18,35,20,34)(21,38,23,40)(24,42,25,41)(27,43,39,44);(7,12)(9,11)(13,17,18,15,19,29,20,30)(14,23,25,39,26,21,24,27)(16,33,35,32,36,37,34,28)(22,38,41,43,31,40,42,44);(6,10)(9,11)(13,18,19,20)(14,24,26,25)(15,29,30,17)(16,34,36,35)(21,39,23,27)(22,41,31,42)(28,37,32,33)(38,43,40,44)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972212);