Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,9)(10,12)(13,14,18,22)(15,23,26,37)(16,28,29,20)(17,31,24,21)(19,32,34,27)(25,39,40,36)(30,35,38,43)(33,41,42,44);(2,3)(6,9)(10,11)(13,15,24,32,18,26,17,27)(14,19,31,37,22,34,21,23)(16,25,38,44,29,40,30,41)(20,33,43,39,28,42,35,36);(1,2,4,3)(5,7,8,12)(6,10,9,11)(13,16)(14,20)(15,25)(17,30)(18,29)(19,33)(21,35)(22,28)(23,36)(24,38)(26,40)(27,41)(31,43)(32,44)(34,42)(37,39);(1,3,4,2)(16,29)(20,28)(25,40)(30,38)(33,42)(35,43)(36,39)(41,44);(15,26)(19,34)(23,37)(25,40)(27,32)(33,42)(36,39)(41,44)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=ghi,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373734);