Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401030 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2541 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14977 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1257 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)(3,8,14,23)(4,9)(5,12)(6,10)(7,11)(13,29,30,21)(15,22,19,24)(16,25,31,32)(17,26)(18,27)(20,28);(1,3,6,15,7,14,5,19)(2,8,12,24,11,23,10,22)(4,16,17,13,18,31,20,30)(9,25,28,21,27,32,26,29);(1,4)(2,9)(3,13)(5,20)(6,17)(7,18)(8,21)(10,26)(11,27)(12,28)(14,30)(15,31)(16,19)(22,25)(23,29)(24,32);(2,10,11,12)(8,22,23,24)(9,26,27,28)(21,25,29,32)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286957, #72287274, #72287308, … |
| maximal normal subgroup of | #72286957, #72287274, #72287308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 315224);