Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2246 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(9,10,12,15,22,24,18,20,14,17,23,19,13,16,21,11);(5,7,8,6)(10,11)(12,21)(13,22)(15,16)(17,20)(18,23)(19,24);(1,2,3,4);(9,11,18,24,13,19,12,10,14,20,21,16,22,15,23,17)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=ehi,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=eh,g^b=gh,h^b=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2177, #72226848, #72226899, … |
| maximal normal subgroup of | D4×#2177, #72226848, #72226899, … |
| maximal quotient of | #72225562, #72226848, C2×#498562, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405758);