Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 | #23848 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,22)(15,21,25,19)(16,20,18,23)(24,27,26,28);(5,7,9,8)(6,10,12,11)(13,15,17,25)(14,19,22,21)(16,26,18,24)(20,27,23,28);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,20)(17,18)(22,23)(24,26)(27,28);(5,8,9,7)(6,10,12,11)(13,15,17,25)(14,21,22,19)(16,26,18,24)(20,28,23,27);(14,22)(15,24)(16,18)(19,27)(21,28)(25,26)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72238725, #72240658, #72249383, … |
| maximal normal subgroup of | #72238725, #72240658, #72249383, … |
| maximal quotient of | #72239716, #72265257, #72307648, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484468);