Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14795 |
| 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 | |
1381 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)(4,17)(5,6)(9,22)(10,21)(11,12)(13,24)(15,20)(23,27)(25,26)(28,30)(31,32);(1,2)(3,13)(4,10)(5,12)(6,8)(7,11)(9,15)(14,27)(16,22)(17,26)(18,21)(19,25)(20,24)(23,30)(28,32)(29,31);(1,3)(2,9)(4,14)(5,15)(6,20)(7,16)(8,13)(10,23)(11,24)(12,22)(17,29)(18,28)(19,30)(21,27)(25,31)(26,32);(1,4,5,19,7,17,6,18)(2,10,12,25,11,26,8,21)(3,14,15,30,16,29,20,28)(9,23,22,31,24,32,13,27)Polycyclic presentation:
2^9|d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72339031, #72339037, #72339042, … |
| maximal normal subgroup of | #72339031, #72339037, #72339042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307251);