Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#1667 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6,9,12)(7,10,8,11)(13,14,18,17,19,22,16,21)(15,23,24,25,26,27,28,20);(1,3,4,2)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,24)(17,27)(18,28)(19,26)(21,23)(22,25);(6,11)(7,8)(10,12)(13,16,19,18)(14,21,22,17)(15,24,26,28)(20,23,25,27);(13,17,18,22,19,21,16,14)(15,25,24,27,26,20,28,23)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=ci,d^b=de,e^b=eh,g^b=gh,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309449, #72309454, #72309525, … |
| maximal normal subgroup of | #72309449, #72309454, #72309525, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 136637);