Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499879 |
| 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 | |
1870 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)(3,12)(4,8)(5,20)(6,11)(7,14)(9,18)(10,21)(13,29)(15,26)(16,28)(17,24)(19,27)(22,31)(23,30)(25,32);(1,3,6,15)(2,7,11,24)(4,17,18,14)(5,13,21,16)(8,12,9,26)(10,22,20,23)(19,31,32,30)(25,28,27,29);(3,13)(4,18)(5,21)(7,22)(10,20)(12,28)(14,31)(15,16)(17,30)(23,24)(25,27)(26,29);(2,8)(4,18)(7,12)(9,11)(10,25)(14,17)(19,32)(20,27)(22,28)(23,29)(24,26)(30,31)Polycyclic presentation:
2^9|b^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fh,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 | #72236766, #72236949, #72237523, … |
| maximal normal subgroup of | #72236949, #72237523, #72312286, … |
| maximal quotient of | #72312378, #72312470, C2×#499882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407078);