Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | #60673 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,28,31,44)(20,25)(21,35)(22,24)(23,30,38,47)(26,33)(27,34)(29,45,32,42)(36,43,48,39)(37,41)(40,46);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23)(20,29)(21,36)(22,31)(24,38)(25,39)(26,42)(27,43)(28,41)(30,46)(32,34)(33,47)(35,44)(37,48)(40,45);(3,9)(6,12)(10,15)(13,16)(18,24)(23,38)(25,27)(26,40)(28,44)(35,41)(39,43)(42,45);(17,20,22,34)(18,25,24,27)(19,29,31,32)(21,33,37,46)(23,39,38,43)(26,41,40,35)(28,45,44,42)(30,36,47,48)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=ci,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72285467, #72285482, #72285524, … |
| maximal normal subgroup of | #72285467, #72285482, #72285524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388509);