Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460622 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16368 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1027 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)(3,7,12,19)(4,14,15,24)(5,10)(6,9)(8,16,22,18)(11,25)(13,21,28,31)(17,23)(20,29)(26,32)(27,30);(1,3)(2,7,9,19)(4,11)(5,13)(6,12)(8,20,22,30)(10,21,23,31)(14,25,24,32)(15,26)(16,27)(17,28)(18,29);(5,17)(7,19)(8,22)(10,23)(11,26)(13,28)(16,18)(20,30);(1,4)(2,8)(3,11)(5,18)(6,15)(7,20)(9,22)(10,24)(12,26)(13,29)(14,23)(16,17)(19,30)(21,32)(25,31)(27,28)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72285055, #72286235, #72286267, … |
| maximal normal subgroup of | #72285055, #72286235, #72286267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367818);