Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1761 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)(6,10)(8,11)(13,14)(15,25)(16,29)(17,23)(18,20)(19,24)(21,26)(22,32)(27,38)(28,37)(30,36)(31,39)(33,35)(34,40)(41,43)(42,44);(5,6,9,10)(7,8,12,11)(14,20)(15,26)(16,30)(17,18)(21,37)(22,29)(23,24)(25,35)(27,41)(31,42)(32,34)(43,44);(2,4)(5,7)(6,11)(8,10)(9,12)(13,15,19,26)(14,21,24,25)(16,31,32,27)(17,33,18,28)(20,35,23,37)(22,39,29,38)(30,41,34,42)(36,44,40,43);(6,10)(7,12)(13,16,19,32)(14,22,24,29)(15,27,26,31)(17,30,18,34)(20,36,23,40)(21,38,25,39)(28,41,33,42)(35,43,37,44);(5,8,9,11)(6,7,10,12)(13,17,19,18)(14,20,24,23)(15,28,26,33)(16,30,32,34)(21,37,25,35)(22,36,29,40)(27,41,31,42)(38,44,39,43)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089339);