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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2054 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)(13,14,20,23,19,25,18,24)(15,26,17,21,29,38,28,39)(16,30,33,41,35,22,34,42)(27,44,36,40,31,43,32,37);(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,21,25,39)(16,31,35,27)(17,20,28,18)(22,40,30,37)(23,26,24,38)(32,33,36,34)(41,43,42,44);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,35)(14,22,25,30)(15,27,29,31)(17,36,28,32)(18,34,20,33)(21,37,39,40)(23,42,24,41)(26,43,38,44);(7,12)(9,11)(13,17,18,15,19,28,20,29)(14,21,24,26,25,39,23,38)(16,32,34,31,35,36,33,27)(22,40,41,44,30,37,42,43);(6,10)(9,11)(13,18,19,20)(14,23,25,24)(15,28,29,17)(16,33,35,34)(21,38,39,26)(22,41,30,42)(27,36,31,32)(37,43,40,44)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,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, 6913952);