Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61266 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18,22,27,24,20,23,35)(19,28,32,29,33,41,31,25)(21,26,38,44,39,34,37,43)(30,42,36,45,46,48,47,40);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,25)(20,29)(21,36)(22,31)(23,32)(24,33)(26,42)(27,41)(28,35)(30,38)(34,48)(37,46)(39,47)(40,44)(43,45);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20,24,18)(19,29,33,25)(21,34,39,26)(22,35,23,27)(28,32,41,31)(30,45,46,40)(36,48,47,42)(37,44,38,43);(4,12)(7,16)(9,11)(13,15)(17,21,23,37,24,39,22,38)(18,26,35,43,20,34,27,44)(19,30,31,47,33,46,32,36)(25,40,41,48,29,45,28,42);(2,8)(3,10)(7,16)(9,11)(17,22,24,23)(18,27,20,35)(19,31,33,32)(21,37,39,38)(25,41,29,28)(26,43,34,44)(30,36,46,47)(40,42,45,48)Polycyclic presentation:
2^9|a^2=fh,c^2=hi,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7012006);