Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467842 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16592 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
842 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)(3,15)(4,20,23,30)(5,24,16,11)(6,13)(7,12)(8,14)(9,17)(10,22,31,26)(18,28)(19,27)(21,29,25,32);(1,3,6,17,8,19,7,18)(2,9,12,15,14,28,13,27)(4,21,26,16,23,25,22,5)(10,29,20,24,31,32,30,11);(1,4,6,22,8,23,7,26)(2,10,12,30,14,31,13,20)(3,16,17,21,19,5,18,25)(9,11,15,32,28,24,27,29);(1,5,8,16)(2,11)(3,4,19,23)(6,21,7,25)(9,10)(12,29)(13,32)(14,24)(15,20)(17,26,18,22)(27,30)(28,31)Polycyclic presentation:
2^9|a^2=fg,b^2=efg,c^2=e,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251686, #72251692, #72251697, … |
| maximal normal subgroup of | #72251686, #72251692, #72251697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375038);