Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #425094 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15328 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,8,6,11)(3,13,14,22,16,24,17,9)(4,18,19,10,21,26,20,23)(15,25,28,27,30,32,29,31);(2,8)(3,14)(5,6)(9,24)(13,22)(15,29)(16,17)(18,26)(19,20)(25,31)(27,32)(28,30);(1,3,5,14,7,16,6,17)(2,9,12,13,8,22,11,24)(4,15,19,28,21,30,20,29)(10,25,26,27,23,32,18,31);(1,4)(2,10)(3,15)(5,19)(6,20)(7,21)(8,23)(9,25)(11,18)(12,26)(13,27)(14,28)(16,30)(17,29)(22,32)(24,31)Polycyclic presentation:
2^9|a^2=ef,c^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251676, #72251685, #72251699, … |
| maximal normal subgroup of | #72251676, #72251685, #72251699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332290);