Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,7,8)(3,4,5,6)(10,13)(11,12)(18,20)(22,24);(1,3)(2,4)(5,7)(6,8)(9,10,14,13)(11,16,12,15)(19,23)(22,24);(1,4,7,6)(2,3,8,5)(9,11,14,12)(10,15,13,16)(17,18,21,20)(19,22,23,24);(1,3,7,5)(2,4,8,6)(9,12,14,11)(10,16,13,15)(17,19,21,23)(18,22,20,24);(1,5,7,3)(2,4,8,6)(9,13,14,10)(11,16,12,15)(17,20,21,18)(19,24,23,22);(1,6,7,4)(2,3,8,5)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72297317, #72297795, C2×#8634851 |
| maximal normal subgroup of | #72297317, #72297795, C2×#8634851 |
| maximal quotient of | #72297439, #72297573, #72297849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8542047);