Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499825 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23900 |
| 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 | |
1930 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)(3,8)(4,7)(5,17)(6,10)(9,18)(11,28)(12,20)(13,25)(14,24)(15,21)(16,27)(19,26)(22,31)(23,30)(29,32);(2,7)(3,11)(5,18)(8,23)(9,27)(10,21)(12,29)(13,30)(14,19)(16,31)(17,22)(20,26)(24,32)(25,28);(1,3)(2,8)(4,13)(5,19)(6,14)(7,20)(9,28)(10,24)(11,18)(12,15)(16,29)(17,26)(21,25)(22,32)(23,27)(30,31);(1,4)(3,12)(5,16)(6,15)(8,24)(11,29)(13,14)(18,31)(19,30)(20,25)(23,32)(26,28)Polycyclic presentation:
2^9|g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gi,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72231746, #72236801, #72236803, … |
| maximal normal subgroup of | #72231746, #72236997, #72237519, … |
| maximal quotient of | C2×#499825, #72280569, #72280900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407021);