Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424595 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2477 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15383 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12,7,13,6,24)(3,14,16,9,17,23,11,5)(4,10,22,28,21,30,20,29)(15,26,27,25,32,31,18,19);(1,3)(2,9)(4,18)(5,13)(6,11)(7,17)(8,16)(10,26)(12,23)(14,24)(15,22)(19,29)(20,32)(21,27)(25,28)(30,31);(1,4)(2,10)(3,15)(5,19)(6,22)(7,21)(8,20)(9,25)(11,27)(12,29)(13,30)(14,31)(16,18)(17,32)(23,26)(24,28);(1,5,7,9)(2,11)(3,12)(4,19,21,25)(6,23,8,14)(10,27)(13,16)(15,29)(17,24)(18,30)(20,31,22,26)(28,32)Polycyclic presentation:
2^9|a^2=ef,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338618, #72338621, #72338635 |
| maximal normal subgroup of | #72338618, #72338621, #72338635 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335603);