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 | #24244 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,20,33,19,24,18,23)(15,25,17,21,28,35,27,36)(16,22,30,37,32,39,31,38)(26,40,29,34,42,44,41,43);(5,6)(7,10)(8,11)(9,12)(13,15,19,28)(14,21,24,36)(16,26,32,42)(17,20,27,18)(22,34,39,43)(23,35,33,25)(29,30,41,31)(37,40,38,44);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16,19,32)(14,22,24,39)(15,26,28,42)(17,29,27,41)(18,31,20,30)(21,34,36,43)(23,38,33,37)(25,40,35,44);(5,8,10,9)(6,11,7,12)(13,17,18,15,19,27,20,28)(14,21,23,25,24,36,33,35)(16,29,31,26,32,41,30,42)(22,34,38,40,39,43,37,44);(5,9,10,8)(6,11,7,12)(13,18,19,20)(14,23,24,33)(15,17,28,27)(16,30,32,31)(21,35,36,25)(22,37,39,38)(26,41,42,29)(34,40,43,44)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gi,d^2=fh,e^2=hi,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=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7106359);