Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468070 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16600 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1118 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)(4,17)(5,12)(6,10)(7,11)(8,15)(9,20)(14,29)(16,27)(18,26)(19,25)(21,23)(22,31)(24,30)(28,32);(1,3)(2,8,11,23)(4,14)(5,15)(6,21)(7,16)(9,24,17,22)(10,13,12,27)(18,30)(19,31)(20,32)(25,28,26,29);(1,4)(2,9,11,17)(3,14)(5,18)(6,19)(7,20)(8,22,23,24)(10,26,12,25)(13,28,27,29)(15,30)(16,32)(21,31);(2,10)(4,18)(5,6)(8,13)(9,17)(11,12)(14,30)(15,21)(19,20)(22,24)(23,27)(31,32)Polycyclic presentation:
2^9|b^2=g,c^2=g,e^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,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72215972, #72216130, #72216203, … |
| maximal normal subgroup of | #72215972, #72216130, #72216203, … |
| maximal quotient of | C2×#468070, #72252122, #72255674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375266);