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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,25)(16,22)(17,23)(18,24)(19,34)(20,29)(21,30)(26,36)(27,41)(28,38)(31,39)(32,37)(33,40)(35,43)(42,44);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,26)(17,32)(18,29)(19,28)(21,35)(22,41)(23,38)(24,25)(27,31)(30,44)(33,43)(34,37)(36,39)(40,42);(5,7)(6,9)(8,12)(10,11)(13,16,18,31)(14,21,24,40)(15,26,29,27)(17,30,19,33)(20,35,25,42)(22,23,39,34)(28,43,32,44)(36,37,41,38);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,27)(17,33)(18,31)(19,30)(20,36)(21,34)(23,40)(24,39)(25,41)(26,29)(28,43)(32,44)(35,38)(37,42);(6,10)(9,11)(13,17,18,19)(14,23,24,34)(15,28,29,32)(16,30,31,33)(20,37,25,38)(21,39,40,22)(26,43,27,44)(35,41,42,36)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903794);