Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470122 |
| 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×#3109 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3366 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)(6,11)(7,12)(8,9)(13,14)(15,19)(16,20)(17,21)(18,22)(23,26)(24,27)(25,28);(2,4)(5,6)(7,8)(9,11)(10,12)(13,15,18,25)(14,19,22,28)(16,24,17,23)(20,27,21,26);(5,7,10,11)(6,8,12,9)(13,16,18,17)(14,20,22,21)(15,23,25,24)(19,26,28,27);(5,8,10,9)(6,7,12,11)(13,17,18,16)(14,21,22,20)(15,24,25,23)(19,27,28,26);(14,22)(19,28)(20,21)(26,27)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283172, #72304664, #72304728, … |
| maximal normal subgroup of | #72283172, #72304664, #72304728, … |
| maximal quotient of | #72304886, #72305067, #72310239, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396854);