Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394131 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,13)(4,18)(5,6)(10,25)(11,12)(14,21)(15,29)(16,17)(19,20)(23,31)(28,32);(1,2)(3,14)(4,10)(5,11)(6,8)(7,12)(9,16)(13,24)(15,28)(17,21)(18,26)(19,22)(20,25)(23,30)(27,32)(29,31);(1,3)(2,9)(4,15)(5,13)(6,16)(7,17)(8,21)(10,23)(11,14)(12,24)(18,27)(19,30)(20,29)(22,31)(25,32)(26,28);(1,4)(2,10)(3,15)(5,18)(6,19)(7,20)(8,22)(9,23)(11,26)(12,25)(13,27)(14,28)(16,30)(17,29)(21,31)(24,32)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72215981, #72215995, … |
| maximal normal subgroup of | #72215958, #72215981, #72215995, … |
| maximal quotient of | C2×#394131, #72252205, #72265220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301327);