Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431710 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23430 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14,17,19,18,22,16,20)(15,21,25,26,23,28,24,27);(1,2)(3,4)(6,9)(10,12)(14,19)(16,17)(20,22)(21,26)(24,25)(27,28);(2,4)(14,20)(16,17)(19,22)(21,27)(24,25)(26,28);(5,7,8,11)(6,10,9,12)(13,15)(14,21)(16,24)(17,25)(18,23)(19,26)(20,27)(22,28);(15,23)(21,28)(24,25)(26,27)Polycyclic presentation:
2^9|a^2=fi,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72239663, #72249598, #72249615 |
| maximal normal subgroup of | #72239663, #72249598, #72249615 |
| maximal quotient of | #72237235, #72237292, #72239613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339815);