Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#99 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(4,5)(8,14)(9,13)(10,11)(15,16)(17,18)(19,22)(20,27,28,23)(21,24)(25,31,32,29)(26,30);(1,2)(3,8)(4,7)(5,13)(6,9)(10,14)(11,16)(12,15)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(1,3,6,12)(2,8,9,15)(4,10,5,11)(7,14,13,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31);(1,4)(3,10)(5,6)(7,13)(11,12)(14,16)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=de,e^b=eh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72268354, #72268359, #72268375, … |
| maximal normal subgroup of | #72268354, #72268359, #72268375, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 127484);