Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1931 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6,10,9)(7,12,8,11)(13,14,18,21)(15,22,25,23)(16,26,20,17)(19,27,28,24);(5,7,10,8)(6,11,9,12)(13,15,17,24,18,25,26,27)(14,19,20,22,21,28,16,23);(2,4)(5,6,10,9)(7,12,8,11)(13,16,17,14,18,20,26,21)(15,23,24,19,25,22,27,28);(5,8,10,7)(6,11,9,12)(13,17,18,26)(14,16,21,20)(15,24,25,27)(19,23,28,22);(5,9,10,6)(7,12,8,11)(15,25)(19,28)(22,23)(24,27)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=f,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307396, #72307414, #72307471, … |
| maximal normal subgroup of | #72307396, #72307414, #72307471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6461907);