Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497333 |
| 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 | #61089 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#576 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1173 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)(7,14)(9,18)(13,24)(15,28,31,27)(16,22)(19,25,32,26)(21,30,23,29);(1,3)(2,7)(4,13)(5,15)(6,16)(8,21)(9,22)(10,23)(11,24)(12,17)(14,18)(19,29)(20,31)(25,28)(26,27)(30,32);(2,8)(5,20)(7,21)(9,25)(10,11)(12,27)(15,31)(17,26)(19,32)(22,28)(23,24)(29,30);(3,13)(4,18)(7,22)(9,17)(12,24)(14,16)(15,29)(19,32)(21,28)(23,27)(25,26)(30,31)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=de,e^b=ei,f^c=fi,g^c=ghi,e^d=ei,g^f=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237519, #72237526, #72237550, … |
| maximal normal subgroup of | #72237519, #72237526, #72237550, … |
| maximal quotient of | #72311990, C2×#497333, #72337427 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404529);