Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1929 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(8,16)(11,17)(15,20)(18,19)(21,22,26,29)(23,30,32,36)(24,28,25,34)(27,31,35,33);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23,25,33,26,32,24,31)(22,27,28,30,29,35,34,36);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,24,26,25)(22,28,29,34)(23,31,32,33)(27,30,35,36);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(23,32)(27,35)(30,36)(31,33)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=g,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307362, #72307702, #72307897, … |
| maximal normal subgroup of | #72307362, #72307702, #72307897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093963);