Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394602 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3003 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,13)(12,16)(14,17)(15,18)(19,20);(6,10)(7,14)(8,16)(11,18)(12,13)(15,20);(5,7,8,14,9,15,16,20)(6,11,12,19,13,18,10,17);(2,4)Polycyclic presentation:
2^9|c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215520, #72216166, #72216203, … |
| maximal normal subgroup of | #72215520, #72216166, #72216203, … |
| maximal quotient of | #72238759, C2×#394602, #72248319, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301798);