Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500755 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2668 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,13)(4,10)(5,8)(6,11)(7,12)(9,15)(14,28)(16,21)(17,26)(18,22)(19,25)(20,24)(23,29)(27,31)(30,32);(2,8)(4,17)(5,6)(9,13)(10,25)(11,12)(14,29)(15,20)(18,19)(21,24)(27,28)(30,31);(1,3)(2,9)(4,14)(5,15)(6,20)(7,16)(8,21)(10,23)(11,13)(12,24)(17,29)(18,30)(19,31)(22,28)(25,32)(26,27);(1,4)(2,10)(3,14)(5,18)(6,17)(7,19)(8,22)(9,23)(11,26)(12,25)(13,27)(15,30)(16,31)(20,29)(21,28)(24,32);(1,5,7,6)(2,8,12,11)(3,15,16,20)(4,18,19,17)(9,21,24,13)(10,22,25,26)(14,30,31,29)(23,28,32,27)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=df,e^b=ei,f^b=fi,g^b=gi,g^c=gi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72216059, #72216095, … |
| maximal normal subgroup of | #72215936, #72216059, #72216095, … |
| maximal quotient of | C2×#7500757, #72238923, #72238925, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407953);