Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993915 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10,14,11)(12,15,13,16)(17,18,19,24,21,20,22,23);(1,3)(2,6)(4,7)(5,8)(10,11)(12,13)(17,19,21,22)(18,23,20,24);(9,11,14,10)(12,15,13,16)(17,20,21,18)(19,24,22,23);(3,8)(6,7)(9,12,14,13)(10,15,11,16);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,15,11,16)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=hi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72251213, #72256835, #72256894, … |
| maximal normal subgroup of | #72251213, #72256835, #72256894, … |
| maximal quotient of | #72264673 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7042268);