Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16266 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23407 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1147 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,16)(5,19)(8,22)(9,25)(12,24)(13,29)(14,15)(17,18)(20,27)(31,32);(1,2,5,10,6,7,19,21)(3,12,11,22,15,8,14,24)(4,9,17,25,18,20,16,27)(13,26,28,31,29,23,30,32);(1,3)(2,8)(4,13)(5,14)(6,15)(7,12)(9,26)(10,22)(11,19)(16,28)(17,30)(18,29)(20,23)(21,24)(25,32)(27,31);(1,4)(2,9)(3,13)(5,17)(6,18)(7,20)(8,23)(10,25)(11,28)(12,26)(14,30)(15,29)(16,19)(21,27)(22,31)(24,32)Polycyclic presentation:
2^9|b^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72338617, #72338624, #72338632 |
| maximal normal subgroup of | #72338617, #72338624, #72338632 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362764);