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 | |
2759 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)(4,18)(5,11)(6,10)(7,12)(8,17)(9,19)(14,25)(15,24)(16,23)(20,27)(21,26)(22,30)(28,31)(29,32);(1,3)(2,8)(4,14)(5,15)(6,17)(7,16)(9,25)(10,23)(11,13)(12,24)(18,28)(19,29)(20,30)(21,31)(22,26)(27,32);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,18)(11,27)(12,26)(13,28)(15,29)(16,30)(17,31)(23,32)(24,25);(3,15)(5,6)(8,23)(10,11)(13,24)(14,29)(16,17)(18,27)(19,21)(22,32)(25,28)(30,31);(3,16)(8,24)(13,23)(14,30)(15,17)(22,25)(28,32)(29,31)Polycyclic presentation:
2^9|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 | #72215912, #72216029, #72216095, … |
| maximal normal subgroup of | #72215912, #72216029, #72216095, … |
| maximal quotient of | C2×#7500755, #72252095, #72255821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407951);