Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573671 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #63003 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,5,7,8)(10,16)(11,12)(13,14)(17,23)(18,19)(20,24);(1,3)(2,5)(4,7)(6,8)(9,10)(11,20)(12,17)(13,16)(14,18)(15,19)(21,24)(22,23);(3,7)(5,8)(9,11)(10,17)(12,15)(13,21)(14,22)(16,23)(18,24)(19,20);(1,3)(2,5)(4,7)(6,8)(9,12,15,11)(10,17,19,20)(13,21,14,22)(16,24,18,23);(3,7)(5,8)(9,13,15,14)(10,18,19,16)(11,21,12,22)(17,24,20,23)Polycyclic presentation:
2^9|a^2=hi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72312145, #72312199, #72312657 |
| maximal normal subgroup of | #72312145, #72312199, #72312657 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6498694);