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 |
| prime radical | #36152 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,14)(13,15)(17,18)(19,22)(20,24)(21,23);(1,3,5,6)(2,4,7,8)(9,10)(11,16)(12,14)(13,15)(17,19,21,24)(18,22,23,20);(1,2)(3,8)(4,6)(5,7)(9,11,13,14)(10,12,15,16)(17,20)(18,19)(21,22)(23,24);(9,10)(11,16)(12,14)(13,15)(17,18)(19,22)(20,24)(21,23);(1,2)(3,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,16);(1,4)(2,6)(3,7)(5,8)(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^9|b^2=hi,c^2=gi,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72233606, #72233690, #72241225, … |
| maximal normal subgroup of | #72233606, #72233690, #72241225, … |
| maximal quotient of | #72303684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7591942);