Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436222 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15551 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12)(3,13)(4,18,21,23)(5,8,6,11)(9,15)(10,20,26,19)(14,28)(16,25)(17,22)(24,31)(27,29)(30,32);(2,8)(4,19)(5,6)(9,13)(11,12)(14,29)(15,16)(18,23)(20,21)(22,25)(24,28)(30,31);(1,3,5,15,7,17,6,16)(2,9,11,13,12,25,8,22)(4,14,19,29,21,31,20,30)(10,24,18,27,26,28,23,32);(1,4,6,20,7,21,5,19)(2,10,11,18,12,26,8,23)(3,14,16,30,17,31,15,29)(9,24,13,27,25,28,22,32)Polycyclic presentation:
2^9|a^2=efg,c^2=e,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251689, #72251692, #72251700, … |
| maximal normal subgroup of | #72251689, #72251692, #72251700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343418);