Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
4114 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)(13,14,20,25)(15,26,16,30)(17,22,32,37)(18,24,19,23)(21,29,36,28)(27,40,31,43)(33,39,34,38)(35,42,44,41);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,20,16)(14,21,25,36)(17,27,32,31)(18,29,19,28)(22,35,37,44)(23,30,24,26)(33,42,34,41)(38,43,39,40);(7,11)(9,12)(13,16,20,15)(14,21,25,36)(17,31,32,27)(18,28,19,29)(22,35,37,44)(23,30,24,26)(33,41,34,42)(38,43,39,40);(7,11)(9,12)(13,17)(14,22)(15,27)(16,31)(18,33)(19,34)(20,32)(21,35)(23,38)(24,39)(25,37)(26,40)(28,41)(29,42)(30,43)(36,44);(6,10)(9,12)(17,32)(22,37)(27,31)(33,34)(35,44)(38,39)(40,43)(41,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377388);