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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1811 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)(15,27,33,21,30,40,29,41)(16,31,34,42,36,22,35,39)(17,37,32,43,38,23,28,44);(5,6)(7,11)(8,10)(9,12)(13,15,19,30)(14,21,26,41)(16,32,36,28)(17,35,38,34)(18,33,20,29)(22,37,31,23)(24,27,25,40)(39,44,42,43);(2,4)(5,7,8,12)(6,9,10,11)(13,16,19,36)(14,22,26,31)(15,28,30,32)(17,29,38,33)(18,35,20,34)(21,23,41,37)(24,39,25,42)(27,43,40,44);(7,12)(9,11)(13,17,20,32,19,38,18,28)(14,23,24,44,26,37,25,43)(15,16,33,34,30,36,29,35)(21,39,40,31,41,42,27,22);(6,10)(9,11)(13,18,19,20)(14,24,26,25)(15,29,30,33)(16,34,36,35)(17,32,38,28)(21,40,41,27)(22,42,31,39)(23,43,37,44)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=fi,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=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972246);