Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #356478 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C42×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | D4×#97 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#347 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,6,13,24,14,7,11,23,18,25,9,28,29)(3,8,16,21,17,27,31,32,15,26,22,4,10,20,30,19);(1,3)(2,8,11,26)(4,18,21,12)(5,22)(6,17)(7,15)(9,20,13,27)(10,25)(14,19,29,32)(16,23)(24,30)(28,31);(2,9,11,13)(4,19,21,32)(5,23)(8,20,26,27)(12,14,18,29)(16,22)(24,28)(30,31);(1,4,5,20,6,19,24,8,7,21,23,27,25,32,28,26)(2,10,12,30,13,3,14,16,11,17,18,31,9,15,29,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,h^2=i,b^a=be,c^a=ch,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=dh,f^c=fi,g^d=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72214039, #72214066, #72214092, … |
| maximal normal subgroup of | #72214039, #72214066, #72214092, … |
| maximal quotient of | #72216232, #72225551, #72225552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400422);