Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C2×#3496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4487 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)(7,9)(8,10)(11,12)(13,14,17,20)(15,23,25,28)(16,22,18,21)(19,26,27,24);(2,4)(5,7,8,11)(6,9,10,12)(13,15,17,25)(14,19,20,27)(16,26,18,24)(21,28,22,23);(6,10)(9,12)(13,14)(15,23)(16,22)(17,20)(18,21)(19,24)(25,28)(26,27);(15,24)(16,18)(19,23)(21,22)(25,26)(27,28);(14,20)(19,27)(21,22)(23,28)Polycyclic presentation:
2^9|a^2=i,b^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249703, #72250121, #72256835, … |
| maximal normal subgroup of | #72249703, #72250121, #72256835, … |
| maximal quotient of | #72249749, #72305836, C2×#6440447 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347643);