Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2676 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15331 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1059 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)(3,13,17,25)(4,18)(5,10)(6,11)(7,12)(8,16,24,14)(9,20)(15,30,31,23)(19,32,29,26)(21,27)(22,28);(1,3)(2,8)(4,19)(5,14)(6,16)(7,17)(9,26)(10,25)(11,13)(12,24)(15,20)(18,30)(21,29)(22,31)(23,27)(28,32);(3,14,17,16)(8,13,24,25)(15,29,31,19)(23,26,30,32);(1,4,5,20,7,21,6,22)(2,9,10,27,12,28,11,18)(3,15,16,19,17,31,14,29)(8,23,13,26,24,30,25,32)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319853, #72338620, #72338623, … |
| maximal normal subgroup of | #72319853, #72338620, #72338623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332450);