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 |
| prime radical | #23802 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,7)(8,11,15,13)(9,14,17,12)(16,19,20,18)(21,22,26,31,27,32,25,30)(23,33,24,29,34,36,35,28);(5,7,10,6)(8,11,15,13)(9,12,17,14)(16,19,20,18)(21,23)(22,28)(24,25)(26,35)(27,34)(29,32)(30,33)(31,36);(1,2,3,4)(5,8,10,15)(6,11,7,13)(9,16,17,20)(12,18,14,19);(8,15)(11,13)(16,20)(18,19)(21,24,25,23,27,35,26,34)(22,29,30,33,32,28,31,36);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(21,25,27,26)(22,30,32,31)(23,24,34,35)(28,33,29,36)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=gh,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308284, #72308650 |
| maximal normal subgroup of | #72308284, #72308650 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105856);