Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418769 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2679 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1959 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(6,12,14,13)(7,15,17,20);(5,6)(7,16)(8,15)(9,13)(10,12)(11,14)(17,18)(19,20);(2,4)(5,7)(6,8)(9,15)(10,20)(11,17)(12,16)(13,18)(14,19);(5,8)(6,7)(9,18)(10,16)(11,19)(12,20)(13,15)(14,17)Polycyclic presentation:
2^9|a^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72258208, #72258209, #72258215, … |
| maximal normal subgroup of | #72258208, #72258209, #72258215, … |
| maximal quotient of | #72258233, #72258245 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325965);