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 | |
6179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(11,16)(14,15)(17,18,21,20)(19,23,24,22);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,19,21,24)(18,22,20,23);(2,4)(6,8)(9,11)(10,14)(12,15)(13,16);(3,7)(6,8)(11,16)(14,15);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,14,16,15)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^9|a^2=g,b^2=g,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72282973, #72304322, #72304372, … |
| maximal normal subgroup of | #72282973, #72304322, #72304372, … |
| maximal quotient of | C2×#7837052, #72336463, #72336966 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7744248);