Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,18,21,24)(16,19,23,26)(22,25,27,28);(6,9)(10,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(1,2,3,4)(15,21)(18,24)(22,27)(25,28);(13,16)(14,19)(15,22)(17,23)(18,25)(20,26)(21,27)(24,28);(16,23)(19,26)(22,27)(25,28);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=i,c^2=h,f^2=g,b^a=bg,c^a=ch,c^b=ci,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72238183, #72238195, #72238222, … |
| maximal normal subgroup of | #72238183, #72238195, #72238222, … |
| maximal quotient of | #72264325, #72264460, #72303236, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536685);