Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
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 | |
11926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(9,10,14,12)(11,16,18,15)(13,20,21,17)(19,22,24,23);(5,6)(7,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,22)(20,23)(21,24);(2,3)(10,12)(11,18)(17,20)(19,24);(1,3,4,2)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24);(9,12,14,10)(11,15,18,16)(13,20,21,17)(19,22,24,23);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72250400, #72306265, #72307394, … |
| maximal normal subgroup of | #72250400, #72306265, #72307394, … |
| maximal quotient of | #72257213, #72257220, #72257916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7585923);