Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4018 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)(7,10)(8,11)(9,12)(13,14,19,22)(15,25,29,36)(16,30,33,38)(17,23,18,24)(20,28,35,27)(21,32,37,31)(26,40,43,34)(39,42,44,41);(2,4)(7,8)(10,11)(13,15,19,29)(14,20,22,35)(16,26,33,43)(17,28,18,27)(21,34,37,40)(23,25,24,36)(30,44,38,39)(31,42,32,41);(5,7,9,8)(6,10,12,11)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,34)(22,37)(23,38)(24,30)(25,39)(27,41)(28,42)(29,43)(35,40)(36,44);(5,8,9,7)(6,11,12,10)(14,22)(20,35)(21,37)(23,24)(25,36)(30,38)(34,40)(39,44);(6,12)(10,11)(13,17,19,18)(14,23,22,24)(15,27,29,28)(16,31,33,32)(20,36,35,25)(21,38,37,30)(26,41,43,42)(34,44,40,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394797);