Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| 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×#3481 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3341 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)(13,14,18,20)(15,22,24,28)(16,21,17,25)(19,26,27,23);(6,10)(8,11)(13,15,18,24)(14,19,20,27)(16,23,17,26)(21,28,25,22);(2,4)(5,6)(7,8)(9,10)(11,12);(5,7,9,12)(6,11,10,8)(14,20)(19,27)(21,25)(22,28);(5,8,9,11)(6,7,10,12)(13,16,18,17)(14,21,20,25)(15,23,24,26)(19,28,27,22)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72307339, #72307419, #72307620, … |
| maximal normal subgroup of | #72307339, #72307419, #72307620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524397);