Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484139 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33605 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,9,8,14,25,23,12,15,10,28,26,31,24,34)(7,16,20,18,21,27,30,33,22,36,19,32,17,35,29,13);(5,7)(6,13,15,33)(8,17)(9,27,28,35)(10,29)(11,30)(12,22)(14,32,31,18)(16,34,36,23)(19,24)(20,25)(21,26);(2,4)(5,8,12,26)(6,14,15,31)(7,17,22,21)(9,23,28,34)(10,24,11,25)(13,32,33,18)(16,35,36,27)(19,30,20,29);(5,9,10,14,8,23,24,15,12,28,11,31,26,34,25,6)(7,18,19,27,21,33,29,36,22,32,20,35,17,13,30,16)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=i,d^2=e,e^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,d^b=de,e^b=eh,g^b=gh,h^b=hi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250700, #72250863, #72250872, … |
| maximal normal subgroup of | #72250700, #72250863, #72250872, … |
| maximal quotient of | #72287271, #72287366, #72287490, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405573);