Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,19)(9,22)(10,14)(11,17)(12,20)(13,25)(16,27)(18,29)(21,30)(23,26)(24,28)(31,32)(33,35)(34,36);(1,3,4,2)(5,7)(6,11)(8,16)(9,23)(10,18)(12,27)(13,29)(14,26)(15,22)(17,25)(19,31)(20,34)(21,33)(24,32)(28,35)(30,36);(11,26)(15,29)(27,35)(31,36);(5,8,10,21)(6,12,14,28)(7,16,18,33)(9,24,25,20)(11,27,26,35)(13,30,22,19)(15,31,29,36)(17,34,23,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286662, #72287028, #72287104, … |
| maximal normal subgroup of | #72286662, #72287028, #72287104, … |
| maximal quotient of | #72314536 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271789);