Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16808 |
| 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 | |
1167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,11,6,9)(3,8,16,23,15,21,13,24)(4,17,18,10,20,25,19,26)(14,28,29,31,30,32,27,22);(1,3)(2,8,11,21)(4,14)(5,13)(6,16)(7,15)(9,23,12,24)(10,22,26,31)(17,28,25,32)(18,27)(19,29)(20,30);(2,9)(3,13)(5,6)(8,21)(10,25)(11,12)(14,29)(15,16)(17,26)(18,19)(22,31)(27,30);(1,4)(2,10)(3,14)(5,18)(6,19)(7,20)(8,22)(9,17)(11,26)(12,25)(13,27)(15,30)(16,29)(21,31)(23,28)(24,32)Polycyclic presentation:
2^9|a^2=f,b^2=egh,f^2=i,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,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237063, #72237073, #72237092, … |
| maximal normal subgroup of | #72237063, #72237073, #72237092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379092);