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 | |
5778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(10,13)(14,16)(17,18)(19,20)(21,22)(23,24);(1,2,5,4)(3,6,7,8)(9,10,12,13)(11,14,15,16);(2,4)(6,8)(18,22)(19,23);(3,7)(6,8)(17,19,21,23)(18,20,22,24);(1,3)(2,6)(4,8)(5,7)(17,20)(18,23)(19,22)(21,24);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|b^2=gi,d^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72331617 |
| maximal normal subgroup of | #72331617 |
| maximal quotient of | #72331690, #72332244, #72332369 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7686845);