Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471461 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16797 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(4,17)(5,6)(10,23)(11,21)(13,22)(14,18)(15,16)(24,32)(27,28)(30,31);(1,2,7,11)(3,13)(4,10,17,28)(5,21,6,8)(9,15)(12,25)(14,29)(16,22)(18,32)(19,23,20,27)(24,30)(26,31);(1,3)(2,9)(4,18)(5,12)(6,15)(7,16)(8,22)(10,26)(11,25)(13,21)(14,19)(17,30)(20,31)(23,32)(24,27)(28,29);(1,4)(2,10)(3,14)(5,20)(6,19)(7,17)(8,23)(9,24)(11,28)(12,18)(13,29)(15,30)(16,31)(21,27)(22,26)(25,32)Polycyclic presentation:
2^9|b^2=egh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237068, #72237073, #72237092, … |
| maximal normal subgroup of | #72237068, #72237073, #72237092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378657);