Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12727 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1782 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C2 | C2×#1782 |
| prime radical | #62815 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #3428 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,9)(5,10,18,7)(6,11)(8,15)(13,25)(14,19,28,22)(16,23,29,20)(17,24)(21,27)(26,30,32,31);(2,7)(5,18)(8,19)(9,20)(10,11)(12,22)(14,28)(16,29)(21,30)(23,24)(25,31)(26,32);(1,3,6,15)(2,8)(4,13,17,27)(5,14,18,28)(7,19)(9,21)(10,22)(11,12)(16,26,29,32)(20,30)(23,31)(24,25);(1,4,6,17)(2,9,11,24)(3,13,15,27)(5,16,18,29)(7,20,10,23)(8,21,12,25)(14,26,28,32)(19,30,22,31)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=ghi,f^c=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261720, #72261805, #72285444, … |
| maximal normal subgroup of | #72261720, #72261805, #72285444, … |
| maximal quotient of | #72285563, #72303832, #72306315 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400659);