Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497339 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #61082 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#577 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,17)(5,10,20,8)(6,11)(7,15)(9,18)(13,22)(14,23)(16,27,28,32)(19,25,31,26)(21,29,24,30);(1,3,6,13)(2,7,11,23)(4,14,18,15)(5,16,20,28)(8,24,10,21)(9,22,17,12)(19,29,31,30)(25,27,26,32);(2,8)(3,13)(5,20)(7,21)(9,25)(10,11)(12,27)(14,15)(17,26)(19,31)(22,32)(23,24);(3,14)(4,18)(7,22)(9,17)(12,23)(13,15)(16,29)(19,31)(21,32)(24,27)(25,26)(28,30)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=de,e^b=ei,f^c=fi,g^c=ghi,e^d=ei,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237521, #72237550, #72312369, … |
| maximal normal subgroup of | #72237521, #72237550, #72312369, … |
| maximal quotient of | #72312373, #72312525, C2×#497340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404536);