Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501108 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24404 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1128 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,9,18,8)(5,10)(6,11)(7,14,22,17)(13,28)(15,29)(16,25)(19,26,32,24)(20,27)(21,30,23,31);(1,3)(2,7)(4,17)(5,13)(6,16)(8,12)(9,25)(10,21)(11,22)(14,18)(15,20)(19,31)(23,27)(24,28)(26,29)(30,32);(3,13)(4,18)(5,20)(7,21)(10,27)(12,28)(14,31)(15,16)(17,30)(22,23)(24,26)(25,29);(2,8)(4,18)(7,12)(9,11)(10,24)(14,17)(19,32)(21,28)(22,25)(23,29)(26,27)(30,31)Polycyclic presentation:
2^9|a^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=eh,g^c=gi,e^d=ei,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72236997, #72237519, #72237526, … |
| maximal normal subgroup of | #72236997, #72237519, #72237526, … |
| maximal quotient of | C2×#501108, #72312018, #72312034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408304);