Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #24425 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1860 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)(15,26)(16,22)(17,23)(18,24)(19,34)(20,25)(21,28)(27,42)(29,38)(30,37)(31,44)(32,39)(33,41)(35,40)(36,43);(5,6)(7,11)(8,10)(9,12)(13,15,20,16)(14,21,25,37)(17,31,33,43)(18,28,19,30)(22,34,26,24)(23,38,41,42)(27,35,29,32)(36,39,44,40);(2,4)(5,7,8,12)(6,9,10,11)(13,16,20,15)(14,22,25,26)(17,29,33,27)(18,28,19,30)(21,34,37,24)(23,38,41,42)(31,32,43,35)(36,40,44,39);(7,12)(9,11)(13,17)(14,23)(15,27)(16,29)(18,32)(19,35)(20,33)(21,36)(22,38)(24,39)(25,41)(26,42)(28,43)(30,31)(34,40)(37,44);(6,10)(9,11)(13,18,20,19)(14,24,25,34)(15,28,16,30)(17,32,33,35)(21,22,37,26)(23,39,41,40)(27,43,29,31)(36,38,44,42)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,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=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6969816);