Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394362 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14716 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2611 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)(3,13)(4,18)(5,12)(6,10)(7,11)(8,17)(9,19)(14,21)(15,23)(16,25)(20,26)(22,24)(27,31)(28,29)(30,32);(1,2)(3,14,16,8)(4,9)(5,10)(6,12)(7,11)(13,17,25,21)(15,28,29,23)(18,19)(20,24)(22,26)(27,30,32,31);(1,3)(2,8)(4,15)(5,21)(6,17)(7,16)(9,23)(10,25)(11,14)(12,13)(18,32)(19,31)(20,29)(22,30)(24,28)(26,27);(1,4)(2,9)(3,15)(5,19)(6,22)(7,20)(8,23)(10,18)(11,24)(12,26)(13,27)(14,28)(16,29)(17,30)(21,31)(25,32)Polycyclic presentation:
2^9|b^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215953, #72216144, #72219683, … |
| maximal normal subgroup of | #72215953, #72216144, #72219683, … |
| maximal quotient of | #72251231, C2×#394362, #72251289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301558);